text: During a long enough time, it is believed that the phase in each coeﬃcient has been ergodic within the interval [0, 2π]. In fact, as analyzed previously, the set deﬁned in Eq. (22), which contains all the possible evolutions commutative with the Hamiltonian18, help to cover the interval [0, 2π] “instantaneously”. Thus, the “volume” of the Hilbert subspace for the free particle is 2πn, (32) { } where n is the number of the non-vanishing coeﬃcients C (0) in Eq. (31). Then the p Boltzmann entropy is given by k ln n + k ln(2π), (33) B B which is independent of energy and space volume, and is con(cid:82)sistent with the deﬁnition in Eq. (24) up to a universal constant. The number n may be identical to the mom(cid:82)entum integral d3p, if all the coeﬃcients { } C (0) are non-vanishing. This is a very special case, since the non-vanishing properties p of those coeﬃcients are uncertain. In order for d3p to be plausible, we should presume a microcanonic(cid:82)al ensemble, in which each particle is assigned to be at some momentum state, with a priori equal probability. This presumed microcanonical ensemble may be expressed | (cid:105)(cid:104) | as ρˆ = 1 d3p p p , with V (p) the volume of the momentum space. Easily to see, V (p) this microcanonical ensemble description breaks the(cid:82)quantum coherence in the quantum { } states, with the quantum information encoded in the coeﬃcients C (0) randomized to p become equal probability [1]. In fact, the integral (d3qd3p)/h3 is only a semiclassical quantity for the “copied” particle in the microcanonical ensemble, since momentum and coordinate operators are noncommutative according to quantum mechanics. Obviously, the Planck’s constant h is inserted because of the noncommutativity of momentum and coordinate operators. But this insertion cannot recover the full quantum description, since the superposition property or quantum coherence for the quantum states of the particle is absent in the integral19. Therefore, the ordinary microstates counting given by ordinary 18Certainly, those evolutions are also commutative with the momentum, since Hˆ = pˆ2/2m in this example. Note that the coordinate operator is not included in the complete set of observables, so our deﬁned entropy should not depend on the space volume. 19One may think that the particle’s states can also be expressed in terms of the eigenstates of particle 20 ##LTLine## ##LTLine##

text: During

text: a

text: long

text: enough

text: time,

text: it

text: is

text: believed

text: that

text: the

text: phase

text: in

text: each

text: coeﬃcient

text: has

text: been

text: ergodic

text: within

text: the

text: interval

text: In

text: fact,

text: as

text: analyzed

text: previously,

text: the

text: set

text: deﬁned

text: in

text: Eq.

text: which

text: contains

text: all

text: the

text: possible

text: evolutions

text: commutative

text: with

text: the

text: help

text: to

text: cover

text: the

text: interval

text: “instantaneously”.

text: Thus,

text: the

text: “volume”

text: of

text: the

text: Hilbert

text: subspace

text: for

text: the

text: free

text: particle

text: is

text: where

text: n

text: is

text: the

text: number

text: of

text: the

text: non-vanishing

text: coeﬃcients

text: C

text: in

text: Eq.

text: Then

text: the

text: p

text: Boltzmann

text: entropy

text: is

text: given

text: by

text: k

text: ln

text: n

text: k

text: B

text: B

text: which

text: is

text: independent

text: of

text: energy

text: and

text: space

text: volume,

text: and

text: is

text: with

text: the

text: deﬁnition

text: in

text: Eq.

text: up

text: to

text: a

text: universal

text: constant.

text: The

text: number

text: n

text: may

text: be

text: identical

text: to

text: the

text: integral

text: if

text: all

text: the

text: coeﬃcients

text: C

text: are

text: non-vanishing.

text: This

text: is

text: a

text: very

text: special

text: case,

text: since

text: the

text: non-vanishing

text: properties

text: p

text: of

text: those

text: coeﬃcients

text: are

text: uncertain.

text: In

text: order

text: for

text: to

text: be

text: plausible,

text: we

text: should

text: presume

text: a

text: ensemble,

text: in

text: which

text: each

text: particle

text: is

text: assigned

text: to

text: be

text: at

text: some

text: momentum

text: state,

text: with

text: a

text: priori

text: equal

text: probability.

text: This

text: presumed

text: microcanonical

text: ensemble

text: may

text: be

text: expressed

text: as

text: ρˆ

text: p

text: p

text: with

text: V

text: (p)

text: the

text: volume

text: of

text: the

text: momentum

text: space.

text: Easily

text: to

text: see,

text: V

text: (p)

text: this

text: microcanonical

text: ensemble

text: description

text: breaks

text: coherence

text: in

text: the

text: quantum

text: states,

text: with

text: the

text: quantum

text: information

text: encoded

text: in

text: the

text: coeﬃcients

text: C

text: randomized

text: to

text: p

text: become

text: equal

text: probability

text: In

text: fact,

text: the

text: integral

text: is

text: only

text: a

text: semiclassical

text: quantity

text: for

text: the

text: “copied”

text: particle

text: in

text: the

text: microcanonical

text: ensemble,

text: since

text: momentum

text: and

text: coordinate

text: operators

text: are

text: noncommutative

text: according

text: to

text: quantum

text: mechanics.

text: Obviously,

text: the

text: Planck’s

text: constant

text: h

text: is

text: inserted

text: because

text: of

text: the

text: noncommutativity

text: of

text: momentum

text: and

text: coordinate

text: operators.

text: But

text: this

text: insertion

text: cannot

text: recover

text: the

text: full

text: quantum

text: description,

text: since

text: the

text: superposition

text: property

text: or

text: quantum

text: coherence

text: for

text: the

text: quantum

text: states

text: of

text: the

text: particle

text: is

text: absent

text: in

text: the

text: Therefore,

text: the

text: ordinary

text: microstates

text: counting

text: given

text: by

text: ordinary

text: those

text: evolutions

text: are

text: also

text: commutative

text: with

text: the

text: momentum,

text: since

text: Hˆ

text: in

text: this

text: example.

text: Note

text: that

text: the

text: coordinate

text: operator

text: is

text: not

text: included

text: in

text: the

text: complete

text: set

text: of

text: observables,

text: so

text: our

text: deﬁned

text: entropy

text: should

text: not

text: depend

text: on

text: the

text: space

text: volume.

text: may

text: think

text: that

text: the

text: particle’s

text: states

text: can

text: also

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: eigenstates

text: of

text: particle

text: ##LTLine##

text: ##LTLine##


text: with some small uncertainty δ. Hence, the deﬁnition k ln Γ with integral over a hypershell B may be treated as the Boltzmann entropy for a canonical ensemble. The canonical ensemble can be eﬀectively described by the concepts of open quantum systems, as will be shown in the next subsection. 3.4 Thermodynamics and statistical mechanics for quantum sys- tems According to the discussions in Sec. , statistical mechanics can be derived by means of open quantum systems. Then, thermodynamics can be derived from statistical mechanics, with temperature emerged as some multiplier [1]. For a quantum closed system with a complete quantum mechanics description, whether thermodynamics could also be derived? If the answer was yes, then we could deﬁne thermodynamics for any quantum system, even for a single quantum particle whose Boltzmann entropy is well given by Eq. (33). In this subsection, we still only consider the energy for simplicity, and try to ﬁnd the concept of temperature for a general quantum system. Thermodynamics is a theory for those macro-quantities or macrostates. According to Sec. , for a quantum closed system, its macrostate is given by two independent quantities (E, S ). Obviously, given only this macrostate, no thermodynamics can be derived Bolt since no temperature can be deﬁned in this case. However, macrostates can change if interactions with some other system are turned on. Assuming two initially independent closed systems a and b, the total Boltzmann entropy for these two uncorrelated systems is H H Sc + Sc = k ln[dim( c) dim( c)], (35) a b B a b according to Eq. (24). Then let a and b interact with each other. Under those interac- tions, the two systems are combined to become a new closed system with a corresponding Boltzmann entropy H Sc = k ln[dim( c )]. (36) ab B ab Then the change of the Boltzmann entropy due to the interactions is given by − H H H ∆S = Sc (Sc + Sc) = k ln[dim( c )/ dim( c) dim( c)], (37) int ab a b B ab a b 22

text: with

label: paragraph

text: some

text: small

text: uncertainty

text: δ.

text: Hence,

text: the

text: deﬁnition

text: k

text: ln

text: Γ

text: with

text: integral

text: over

text: a

text: hypershell

text: B

text: may

text: be

text: treated

text: as

text: the

text: Boltzmann

text: entropy

text: for

text: a

text: canonical

text: ensemble.

text: The

text: canonical

text: ensemble

text: can

text: be

text: eﬀectively

text: described

text: by

text: the

text: concepts

text: of

text: open

text: quantum

text: systems,

text: as

text: will

text: be

text: shown

text: in

text: the

text: next

text: subsection.

label: section

text: Thermodynamics

text: and

text: statistical

text: mechanics

text: for

text: quantum

text: sys-

text: tems

text: According

text: to

text: the

text: discussions

text: in

text: Sec.

text: statistical

text: mechanics

text: can

text: be

text: derived

text: by

text: means

text: of

text: open

text: quantum

text: systems.

text: Then,

text: thermodynamics

text: can

text: be

text: derived

text: from

text: statistical

text: mechanics,

text: with

text: temperature

text: emerged

text: as

text: some

text: multiplier

text: For

text: a

text: quantum

text: closed

text: system

text: with

text: a

text: complete

text: quantum

text: mechanics

text: description,

text: whether

text: thermodynamics

text: could

text: also

text: be

text: derived?

text: If

text: the

text: answer

text: was

text: yes,

text: then

text: we

text: could

text: deﬁne

text: thermodynamics

text: for

text: any

text: quantum

text: system,

text: even

text: for

text: a

text: single

text: quantum

text: particle

text: whose

text: Boltzmann

text: entropy

text: is

text: well

text: given

text: by

text: Eq.

text: In

text: this

text: subsection,

text: we

text: still

text: only

text: consider

text: the

text: energy

text: for

text: simplicity,

text: and

text: try

text: to

text: ﬁnd

text: the

text: concept

text: of

text: temperature

text: for

text: a

text: general

text: quantum

text: system.

text: Thermodynamics

text: is

text: a

text: theory

text: for

text: those

text: macro-quantities

text: or

text: macrostates.

text: According

text: to

text: Sec.

text: for

text: a

text: quantum

text: closed

text: system,

text: its

text: macrostate

text: is

text: given

text: by

text: two

text: independent

text: quantities

text: (E,

text: S

text: Obviously,

text: given

text: only

text: this

text: macrostate,

text: no

text: thermodynamics

text: can

text: be

text: derived

text: Bolt

text: since

text: no

text: temperature

text: can

text: be

text: deﬁned

text: in

text: this

text: case.

text: However,

text: macrostates

text: can

text: change

text: if

text: interactions

text: with

text: some

text: other

text: system

text: are

text: turned

text: on.

text: Assuming

text: two

text: initially

text: independent

text: closed

text: systems

text: a

text: and

text: b,

text: the

text: total

text: Boltzmann

text: entropy

text: for

text: these

text: two

text: uncorrelated

text: systems

text: is

text: H

text: H

text: Sc

text: Sc

text: k

text: ln[dim(

text: c)

text: dim(

text: c)],

text: a

text: b

text: B

text: a

text: b

text: according

text: to

text: Eq.

text: Then

text: let

text: a

text: and

text: b

text: interact

text: with

text: each

text: other.

text: Under

text: those

text: interac-

text: tions,

text: the

text: two

text: systems

text: are

text: combined

text: to

text: become

text: a

text: new

text: closed

text: system

text: with

text: a

text: corresponding

text: Boltzmann

text: entropy

text: H

text: Sc

text: k

text: ln[dim(

text: c

text: ab

text: B

text: ab

text: Then

text: the

text: change

text: of

text: the

text: Boltzmann

text: entropy

text: due

text: to

text: the

text: interactions

text: is

text: given

text: by

text: H

text: H

text: H

text: ∆S

text: Sc

text: (Sc

text: Sc)

text: k

text: ln[dim(

text: c

text: dim(

text: c)

text: dim(

text: c)],

text: int

text: ab

text: a

text: b

text: B

text: ab

text: a

text: b


text: − − (cid:39) dition for each subsystem only gives its own temperature, such as δ(∆S T 1∆E ) 0, − − (cid:39) a a a and δ(∆S T 1∆E ) 0. Without the “detailed balance” conditions, the two tempera- b b b tures cannot be identiﬁed, and the thermal equilibrium cannot be established eﬀectively. Furthermore, for a more larger closed system composed of more than two (macroscopic) components, the “detailed balance” conditions between any two components may be pro- posed approximately, provided the stability conditions for the rest components could be presupposed appropriately. Certainly, this depends on the details of the relevant interac- tions, and needs further investigations. Here adds some notes. Although the above macro-quantities are usually deﬁned as quantum averages like the one in Eq. (20), their relations are also applicable classically. For example, the ﬁrst law has the similar form as that of ordinary thermodynamics. In this sense, we can conclude that thermodynamics and statistical mechanics indeed can be derived from quantum mechanics step by step. Further, the thermal meaning of the temperatures T , T and T can also be seen in the following way. The relations in Eq. (42) a b are unchanged by adding extra terms 0 = ∆Q + ∆Q and 0 = ∆SQ + ∆SQ, which are a b a b contributions of the classical heat. This means that there are classical heat and entropy ﬂows between a and b. Under this circumstance, the thermal temperature between them is given by T = ∆Q /∆SQ = ∆Q /∆SQ = T . This can also be treated as the deﬁnition a a a b b b of thermal temperature in a measurement sense, if system b is some thermometer whose perturbations on the studied system can be neglected. These conﬁrm that T deﬁned in Eq. (39) indeed can be treated as a thermal temperature. The distinction is that T or a T has more practical meaning, because only local measurements as in Eq. (40) have been b performed. But T also applies to a general process even the involved interactions are strong, while T or T is meaningful only if the interactions are weak enough so that the a b change ∆E (∆S ) can even be replaced by a diﬀerential dE (dS ), leading to the ﬁrst law a a a a dE = T dS in the diﬀerential form. a a a The above derived thermodynamic rules are almost formally similar to the ordinary one. In particular, the “detailed balance” conditions in Eq. (46) between systems a and b should also be proper for ordinary thermodynamics, with the Boltzmann entropies given by the approximation of the corresponding entanglement entropies in the stable limit. This is because the “detailed balance” conditions are also derived in the stable limit, according to some stability requirement that the observed macro-quantities for a closed system should be almost invariant in its (internal) thermal equilibrium. The physical meanings for those “detailed balance” conditions are easy to understand. The ﬁrst energy condition says that the total energy of the entire closed system is (almost) stable, while 28

text: dition

text: for

text: each

text: subsystem

text: only

text: gives

text: its

text: own

text: temperature,

text: such

text: as

text: δ(∆S

text: T

text: a

text: a

text: a

text: and

text: δ(∆S

text: T

text: Without

text: the

text: “detailed

text: balance”

text: conditions,

text: the

text: two

text: tempera-

text: b

text: b

text: b

text: tures

text: cannot

text: be

text: identiﬁed,

text: and

text: the

text: thermal

text: equilibrium

text: cannot

text: be

text: established

text: eﬀectively.

text: Furthermore,

text: for

text: a

text: more

text: larger

text: closed

text: system

text: composed

text: of

text: more

text: than

text: two

text: (macroscopic)

text: components,

text: the

text: “detailed

text: balance”

text: conditions

text: between

text: any

text: two

text: components

text: may

text: be

text: pro-

text: posed

text: approximately,

text: provided

text: the

text: stability

text: conditions

text: for

text: the

text: rest

text: components

text: could

text: be

text: presupposed

text: appropriately.

text: Certainly,

text: this

text: depends

text: on

text: the

text: details

text: of

text: the

text: relevant

text: interac-

text: tions,

text: and

text: needs

text: further

text: investigations.

text: Here

text: adds

text: some

text: notes.

text: Although

text: the

text: above

text: macro-quantities

text: are

text: usually

text: deﬁned

text: as

text: quantum

text: averages

text: like

text: the

text: one

text: in

text: Eq.

text: their

text: relations

text: are

text: also

text: applicable

text: classically.

text: For

text: example,

text: the

text: ﬁrst

text: law

text: has

text: the

text: similar

text: form

text: as

text: that

text: of

text: ordinary

text: thermodynamics.

text: In

text: this

text: sense,

text: we

text: can

text: conclude

text: that

text: thermodynamics

text: and

text: statistical

text: mechanics

text: indeed

text: can

text: be

text: derived

text: from

text: quantum

text: mechanics

text: step

text: by

text: step.

text: Further,

text: the

text: thermal

text: meaning

text: of

text: the

text: temperatures

text: T

text: T

text: and

text: T

text: can

text: also

text: be

text: seen

text: in

text: the

text: following

text: way.

text: The

text: relations

text: in

text: Eq.

text: a

text: b

text: are

text: unchanged

text: by

text: adding

text: extra

text: terms

text: ∆Q

text: ∆Q

text: and

text: ∆SQ

text: ∆SQ,

text: which

text: are

text: a

text: b

text: a

text: b

text: contributions

text: of

text: the

text: classical

text: heat.

text: This

text: means

text: that

text: there

text: are

text: classical

text: heat

text: and

text: entropy

text: ﬂows

text: between

text: a

text: and

text: b.

text: Under

text: this

text: circumstance,

text: the

text: thermal

text: temperature

text: between

text: them

text: is

text: given

text: by

text: T

text: ∆Q

text: /∆SQ

text: ∆Q

text: /∆SQ

text: T

text: This

text: can

text: also

text: be

text: treated

text: as

text: the

text: deﬁnition

text: a

text: a

text: a

text: b

text: b

text: b

text: of

text: thermal

text: temperature

text: in

text: a

text: measurement

text: sense,

text: if

text: system

text: b

text: is

text: some

text: thermometer

text: whose

text: perturbations

text: on

text: the

text: studied

text: system

text: can

text: be

text: neglected.

text: These

text: conﬁrm

text: that

text: T

text: deﬁned

text: in

text: Eq.

text: indeed

text: can

text: be

text: treated

text: as

text: a

text: thermal

text: temperature.

text: The

text: distinction

text: is

text: that

text: T

text: or

text: a

text: T

text: has

text: more

text: practical

text: meaning,

text: because

text: only

text: local

text: measurements

text: as

text: in

text: Eq.

text: have

text: been

text: b

text: performed.

text: But

text: T

text: also

text: applies

text: to

text: a

text: general

text: process

text: even

text: the

text: involved

text: interactions

text: are

text: strong,

text: while

text: T

text: or

text: T

text: is

text: meaningful

text: only

text: if

text: the

text: interactions

text: are

text: weak

text: enough

text: so

text: that

text: the

text: a

text: b

text: change

text: ∆E

text: (∆S

text: can

text: even

text: be

text: replaced

text: by

text: a

text: diﬀerential

text: dE

text: (dS

text: leading

text: to

text: the

text: ﬁrst

text: law

text: a

text: a

text: a

text: a

text: dE

text: T

text: dS

text: in

text: the

text: diﬀerential

text: form.

text: a

text: a

text: a

text: The

text: above

text: derived

text: thermodynamic

text: rules

text: are

text: almost

text: formally

text: similar

text: to

text: the

text: ordinary

text: one.

text: In

text: particular,

text: the

text: “detailed

text: balance”

text: conditions

text: in

text: Eq.

text: between

text: systems

text: a

text: and

text: b

text: should

text: also

text: be

text: proper

text: for

text: ordinary

text: thermodynamics,

text: with

text: the

text: Boltzmann

text: entropies

text: given

text: by

text: the

text: approximation

text: of

text: the

text: corresponding

text: entanglement

text: entropies

text: in

text: the

text: stable

text: limit.

text: This

text: is

text: because

text: the

text: “detailed

text: balance”

text: conditions

text: are

text: also

text: derived

text: in

text: the

text: stable

text: limit,

text: according

text: to

text: some

text: stability

text: requirement

text: that

text: the

text: observed

text: macro-quantities

text: for

text: a

text: closed

text: system

text: should

text: be

text: almost

text: invariant

text: in

text: its

text: (internal)

text: thermal

text: equilibrium.

text: The

text: physical

text: meanings

text: for

text: those

text: “detailed

text: balance”

text: conditions

text: are

text: easy

text: to

text: understand.

text: The

text: ﬁrst

text: energy

text: condition

text: says

text: that

text: the

text: total

text: energy

text: of

text: the

text: entire

text: closed

text: system

text: is

text: (almost)

text: stable,

text: while


text: chanics, so that the ﬁctitious information from the concept of ensemble can be removed. We show that the (grand) canonical ensemble theory of statistical mechanics can be de- rived eﬀectively by means of open quantum systems. In particular, the most probable distribution for statistical mechanics will correspond to some stable one for the studied open quantum system. In this way, the Boltzmann entropy for the (grand) canonical en- semble can be treated as an approximation of some entanglement entropy in the stable limit. Furthermore, a new deﬁnition of Boltzmann entropy for a quantum closed system is also given, which counts microstates in a way consistent with the superposition principle of quantum mechanics. In particular, that new Boltzmann entropy is a constant that doesn’t depend on the system’s energy and space volume. In fact, it is identical to the maximum von Neumann’s entropy related to the density matrixes for some measurements, thus it depends only on the dimension of the system’s relevant Hilbert subspace. This paper is organized as follows. In Sec. , some basic properties of entanglement entropy or von Neumann entropy are brieﬂy mentioned for comparing to the Boltzmann entropy. In Sec. , some general investigations are given to show that the (grand) canonical ensemble theory of statistical mechanics can be derived eﬀectively by means of open quan- tum systems. While in Sec. , a new Boltzmann entropy is deﬁned for a quantum closed system, which is a constant independent of the system’s energy and space volume. Some more details for thermodynamics of quantum systems, especially the thermal equilibrium between two macroscopic subsystems are given in Sec. . In addition, two simple examples are also analyzed in Sec. and the Appendix . 2 A Brief Introduction to Entanglement Entropy In this intermediate section, a brief introduction to entanglement entropy is provided to compare with the Boltzmann entropy. An entanglement entropy is deﬁned as − S = T r(ρˆ ln ρˆ ), (2) en ext ext which is constructed from some reduced density matrix. A reduced density matrix could occur only if the relevant system contained two or more than two independent degrees of freedom2. According to the standard results of quantum mechanics, a complete collection 2Specially, when making a measurement on a single quantum degree of freedom, the apparatus should 3 ##LTLine##

text: chanics,

text: so

text: that

text: the

text: ﬁctitious

text: information

text: from

text: the

text: concept

text: of

text: ensemble

text: can

text: be

text: removed.

text: We

text: show

text: that

text: the

text: (grand)

text: canonical

text: ensemble

text: theory

text: of

text: statistical

text: mechanics

text: can

text: be

text: de-

text: rived

text: eﬀectively

text: by

text: means

text: of

text: open

text: quantum

text: systems.

text: In

text: particular,

text: the

text: most

text: probable

text: distribution

text: for

text: statistical

text: mechanics

text: will

text: correspond

text: to

text: some

text: stable

text: one

text: for

text: the

text: studied

text: open

text: quantum

text: system.

text: In

text: this

text: way,

text: the

text: Boltzmann

text: entropy

text: for

text: the

text: (grand)

text: canonical

text: en-

text: semble

text: can

text: be

text: treated

text: as

text: an

text: approximation

text: of

text: some

text: entanglement

text: entropy

text: in

text: the

text: stable

text: limit.

text: Furthermore,

text: a

text: new

text: deﬁnition

text: of

text: Boltzmann

text: entropy

text: for

text: a

text: quantum

text: closed

text: system

text: is

text: also

text: given,

text: which

text: counts

text: microstates

text: in

text: a

text: way

text: consistent

text: with

text: the

text: superposition

text: principle

text: of

text: quantum

text: mechanics.

text: In

text: particular,

text: that

text: new

text: Boltzmann

text: entropy

text: is

text: a

text: constant

text: that

text: doesn’t

text: depend

text: on

text: the

text: system’s

text: energy

text: and

text: space

text: volume.

text: In

text: fact,

text: it

text: is

text: identical

text: to

text: the

text: maximum

text: von

text: Neumann’s

text: entropy

text: related

text: to

text: the

text: density

text: matrixes

text: for

text: some

text: measurements,

text: thus

text: it

text: depends

text: only

text: on

text: the

text: dimension

text: of

text: the

text: system’s

text: relevant

text: Hilbert

text: subspace.

text: This

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: some

text: basic

text: properties

text: of

text: entanglement

text: entropy

text: or

text: von

text: Neumann

text: entropy

text: are

text: brieﬂy

text: mentioned

text: for

text: comparing

text: to

text: the

text: Boltzmann

text: entropy.

text: In

text: Sec.

text: some

text: general

text: investigations

text: are

text: given

text: to

text: show

text: that

text: the

text: (grand)

text: canonical

text: ensemble

text: theory

text: of

text: statistical

text: mechanics

text: can

text: be

text: derived

text: eﬀectively

text: by

text: means

text: of

text: open

text: quan-

text: tum

text: systems.

text: While

text: in

text: Sec.

text: a

text: new

text: Boltzmann

text: entropy

text: is

text: deﬁned

text: for

text: a

text: quantum

text: closed

text: system,

text: which

text: is

text: a

text: constant

text: independent

text: of

text: the

text: system’s

text: energy

text: and

text: space

text: volume.

text: Some

text: more

text: details

text: for

text: thermodynamics

text: of

text: quantum

text: systems,

text: especially

text: the

text: thermal

text: equilibrium

text: between

text: two

text: macroscopic

text: subsystems

text: are

text: given

text: in

text: Sec.

text: In

text: addition,

text: two

text: simple

text: examples

text: are

text: also

text: analyzed

text: in

text: Sec.

text: and

text: the

text: Appendix

text: A

text: Brief

text: Introduction

text: to

text: Entanglement

text: Entropy

text: In

text: this

text: intermediate

text: section,

text: a

text: brief

text: introduction

text: to

text: entanglement

text: entropy

text: is

text: provided

text: to

text: compare

text: with

text: the

text: Boltzmann

text: entropy.

text: An

text: entanglement

text: entropy

text: is

text: deﬁned

text: as

text: S

text: T

text: r(ρˆ

text: ln

text: ρˆ

text: en

text: ext

text: ext

text: which

text: is

text: constructed

text: from

text: some

text: reduced

text: density

text: matrix.

text: A

text: reduced

text: density

text: matrix

text: could

text: occur

text: only

text: if

text: the

text: relevant

text: system

text: contained

text: two

text: or

text: more

text: than

text: two

text: independent

text: degrees

text: of

text: According

text: to

text: the

text: standard

text: results

text: of

text: quantum

text: mechanics,

text: a

text: complete

text: collection

text: when

text: making

text: a

text: measurement

text: on

text: a

text: single

text: quantum

text: degree

text: of

text: freedom,

text: the

text: apparatus

text: should

text: ##LTLine##


text:   (cid:88)   ≈ { } S k ln dim(G n ) , (59) Bolt B p i { } ni which is the entropy (increment) for the system induced by the interactions. This is one part of of the full change k ln[dim(G )], just like the one in Eq. (49). The next step is B Uˆint to ﬁnd the most probable distribution that maximizes the Boltzmann entropy (increment) together with the some restrictions on the energy and particle number, just like Eq. (50), giving the familiar distribution for the indistinguishable quantum particles33. The above derivation is diﬀerent from the one in a standard textbook [1], where the { } latter case with (sub-)levels (cid:15) (λ) is treated as a microcanonical ensemble for a closed i system. However, if treated as a (quantum) closed system, it can be well described by { } ˆ quantum mechanics in Fock space representation, with a Hamiltonian H nˆ (λ) . In par- i ticular, the permutational properties can be well described by the commutative relations of † the creators and annihilators [aˆ (λi ), aˆ (λj )]∓ = δ δ . In this sense, the enumeration of i α j β ij αβ microstates due to the permutations in the textbook’s derivation is artiﬁcial due to a priori { } coarse grained description of the (sub-)levels (cid:15) (λ) . In our derivation, the Boltzmann en- i tropy related to the permutations comes from the weak interactions with an environment, with which the system is in equilibrium according to the (stability) conditions in Eq. (52). This gives a grand canonical ensemble for the system open to an environment, similarly ˆ ˆ for canonical ensemble if the condition for particle number is replaced by [U , N] = 0. int Acknowledgments This work is supported by the NNSF of China, Grant No. 11375150. References [1] R.K. Pathria and P.D. Beale, Statistical Mechanics, Third edition, Elsevier (2011). [2] J. Preskill, Physics 229: Advanced Mathematical Methods of Physics, Quantum Com- putation and Information, California Institute of Technology (1998). [3] Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum In- formation, Cambridge University Press (2010). 33Note that, although the energy levels are split into sub-levels by the interactions, the macro-observables in Eq. (51) are almost unchanged. Thus, the energy and particle number can still be expressed approxi- { } mately in terms of the distribution n . i 32 ##LTLine##

label: equation

text: S

text: k

text: ln

text: dim(G

text: n

text: Bolt

text: B

text: p

text: i

text: ni

text: which

text: is

text: the

text: entropy

text: (increment)

text: for

text: the

text: system

text: induced

text: by

text: the

text: interactions.

text: This

text: is

text: one

text: part

text: of

text: of

text: the

text: full

text: change

text: k

text: ln[dim(G

text: just

text: like

text: the

text: one

text: in

text: Eq.

text: The

text: next

text: step

text: is

text: B

text: Uˆint

text: to

text: ﬁnd

text: the

text: most

text: probable

text: distribution

text: that

text: maximizes

text: the

text: Boltzmann

text: entropy

text: (increment)

text: together

text: with

text: the

text: some

text: restrictions

text: on

text: the

text: energy

text: and

text: particle

text: number,

text: just

text: like

text: Eq.

text: giving

text: the

text: familiar

text: distribution

text: for

text: the

text: indistinguishable

text: quantum

text: The

text: above

text: derivation

text: is

text: diﬀerent

text: from

text: the

text: one

text: in

text: a

text: standard

text: textbook

text: where

text: the

text: latter

text: case

text: with

text: (sub-)levels

text: (λ)

text: is

text: treated

text: as

text: a

text: microcanonical

text: ensemble

text: for

text: a

text: closed

text: i

text: system.

text: However,

text: if

text: treated

text: as

text: a

text: (quantum)

text: closed

text: system,

text: it

text: can

text: be

text: well

text: described

text: by

text: ˆ

text: quantum

text: mechanics

text: in

text: Fock

text: space

text: representation,

text: with

text: a

text: Hamiltonian

text: H

text: nˆ

text: (λ)

text: In

text: par-

text: i

text: ticular,

text: the

text: permutational

text: properties

text: can

text: be

text: well

text: described

text: by

text: the

text: commutative

text: relations

text: of

text: the

text: creators

text: and

text: annihilators

text: [aˆ

text: (λi

text: aˆ

text: (λj

text: δ

text: δ

text: In

text: this

text: sense,

text: the

text: enumeration

text: of

text: i

text: α

text: j

text: β

text: ij

text: αβ

text: microstates

text: due

text: to

text: the

text: permutations

text: in

text: the

text: textbook’s

text: derivation

text: is

text: artiﬁcial

text: due

text: to

text: a

text: priori

text: coarse

text: grained

text: description

text: of

text: the

text: (sub-)levels

text: (λ)

text: In

text: our

text: derivation,

text: the

text: Boltzmann

text: en-

text: i

text: tropy

text: related

text: to

text: the

text: permutations

text: comes

text: from

text: the

text: weak

text: interactions

text: with

text: an

text: environment,

text: with

text: which

text: the

text: system

text: is

text: in

text: equilibrium

text: according

text: to

text: the

text: (stability)

text: conditions

text: in

text: Eq.

text: This

text: gives

text: a

text: grand

text: canonical

text: ensemble

text: for

text: the

text: system

text: open

text: to

text: an

text: environment,

text: similarly

text: ˆ

text: ˆ

text: for

text: canonical

text: ensemble

text: if

text: the

text: condition

text: for

text: particle

text: number

text: is

text: replaced

text: by

text: [U

text: N]

text: int

text: Acknowledgments

text: This

text: work

text: is

text: supported

text: by

text: the

text: NNSF

text: of

text: China,

text: Grant

text: No.

text: References

text: R.K.

text: Pathria

text: and

text: P.D.

text: Beale,

text: Statistical

text: Mechanics,

text: Third

text: edition,

text: Elsevier

text: J.

text: Preskill,

text: Physics

text: Advanced

text: Mathematical

text: Methods

text: of

text: Physics,

text: Quantum

text: Com-

text: putation

text: and

text: Information,

text: California

text: Institute

text: of

text: Technology

text: Michael

text: A.

text: Nielsen

text: and

text: Isaac

text: L.

text: Chuang,

text: Quantum

text: Computation

text: and

text: Quantum

text: In-

text: formation,

text: Cambridge

text: University

text: Press

text: that,

text: although

text: the

text: energy

text: levels

text: are

text: split

text: into

text: sub-levels

text: by

text: the

text: interactions,

text: the

text: macro-observables

text: in

text: Eq.

text: are

text: almost

text: unchanged.

text: Thus,

text: the

text: energy

text: and

text: particle

text: number

text: can

text: still

text: be

text: expressed

text: approxi-

text: mately

text: in

text: terms

text: of

text: the

text: distribution

text: n

text: i

text: ##LTLine##


text: observables are diﬃcult to be constructed. Usually, the complete set of observables may { · · · } { · · · } ˆ ˆ ˆ ˆ ˆ ˆ be given by H, N, O , O , with O , O , denoted as the other inaccessible ob- 1 2 1 2 servables. The system’s states are given by superpositions of basis states or eigenstates (cid:88) for those complete set of observables, i.e. | (cid:105) | · · · (cid:105) ψ(t) = C ···(t) E , N , , (6) r,s, r s ··· r,s, where the quantum numbers for those inaccessible observables are ignored for simplic- ity. Then we turn on some interactions from an environment. It’s assumed that the inter- { · · · } ˆ ˆ ˆ ˆ actions are weak enough so that the the complete set of observables H, N, O , O , 1 2 (cid:88) can still be used approximately. Then the state of the combined system-environment is | (cid:105) | · · · (cid:105)| (cid:105) Ψ(t) = D ··· (t) E , N , χ , (7) r,s, ;χ r s E ··· r,s, ;χ | (cid:105) where χ is the state of the environment, with χ denoted as the corresponding quan- E tum numbers for short. Focusing only on the energy and particle number, after partially tracing over the environment and those inaccessible degrees of freedom, we will obtain a (cid:88) reduced density matrix | (cid:105)(cid:104) | ρˆ (t) = ρ (t) E , N E , N . (8) red r,s;r˜,s˜ r s r˜ s˜ r,s;r˜,s˜ For complex macroscopic systems, the involved interactions are so complicated that the oﬀ-diagonal terms in Eq. (8) cannot be studied well. Fortunately, the oﬀ-diagonal terms ˆ ˆ do not contribute to the energy E = T r(Hρˆ (t)) and particle number N = T r(Nρˆ (t)). red red Under this circumstance, we can make an eﬀective reduction → ≡ ρ (t) P (t) ρ (t), (9) r,s;r˜,s˜ r,s r,s;r,s {| (cid:105)} obtaining some probability distribution of the collection of states E , N . This pro- r s vides a “quasi-ensemble” under some (super-operator) evolution [6] resulted from those interactions. Certainly, this “quasi-ensemble” is not the presumed ensemble used in sta- tistical mechanics. The main diﬀerence is that, in the presumed ensemble the member systems are assumed to be uncorrelated or independent [1]. While the members in the 6

text: observables

text: are

text: diﬃcult

text: to

text: be

text: constructed.

text: Usually,

text: the

text: complete

text: set

text: of

text: observables

text: may

text: ˆ

text: ˆ

text: ˆ

text: ˆ

text: ˆ

text: ˆ

text: be

text: given

text: by

text: H,

text: N,

text: O

text: O

text: with

text: O

text: O

text: denoted

text: as

text: the

text: other

text: inaccessible

text: ob-

text: servables.

text: The

text: system’s

text: states

text: are

text: given

text: by

text: superpositions

text: of

text: basis

text: states

text: or

text: eigenstates

text: for

text: those

text: complete

text: set

text: of

text: observables,

text: i.e.

text: ψ(t)

text: C

text: ···(t)

text: E

text: N

text: r,s,

text: r

text: s

text: r,s,

text: where

text: the

text: quantum

text: numbers

text: for

text: those

text: inaccessible

text: observables

text: are

text: ignored

text: for

text: simplic-

text: ity.

text: Then

text: we

text: turn

text: on

text: some

text: interactions

text: from

text: an

text: environment.

text: It’s

text: assumed

text: that

text: the

text: inter-

text: ˆ

text: ˆ

text: ˆ

text: ˆ

text: actions

text: are

text: weak

text: enough

text: so

text: that

text: the

text: the

text: complete

text: set

text: of

text: observables

text: H,

text: N,

text: O

text: O

text: can

text: still

text: be

text: used

text: approximately.

text: Then

text: the

text: state

text: of

text: the

text: combined

text: system-environment

text: is

text: Ψ(t)

text: D

text: (t)

text: E

text: N

text: χ

text: r,s,

text: ;χ

text: r

text: s

text: E

text: r,s,

text: ;χ

text: where

text: χ

text: is

text: the

text: state

text: of

text: the

text: environment,

text: with

text: χ

text: denoted

text: as

text: the

text: corresponding

text: quan-

text: E

text: tum

text: numbers

text: for

text: short.

text: Focusing

text: only

text: on

text: the

text: energy

text: and

text: particle

text: number,

text: after

text: partially

text: tracing

text: over

text: the

text: environment

text: and

text: those

text: inaccessible

text: degrees

text: of

text: freedom,

text: we

text: will

text: obtain

text: a

text: reduced

text: density

text: matrix

text: ρˆ

text: (t)

text: ρ

text: (t)

text: E

text: N

text: E

text: N

text: red

text: r,s;r˜,s˜

text: r

text: s

text: r˜

text: s˜

text: r,s;r˜,s˜

text: For

text: complex

text: macroscopic

text: systems,

text: the

text: involved

text: interactions

text: are

text: so

text: complicated

text: that

text: the

text: oﬀ-diagonal

text: terms

text: in

text: Eq.

text: cannot

text: be

text: studied

text: well.

text: Fortunately,

text: the

text: oﬀ-diagonal

text: terms

text: ˆ

text: ˆ

text: do

text: not

text: contribute

text: to

text: the

text: energy

text: E

text: T

text: r(Hρˆ

text: (t))

text: and

text: particle

text: number

text: N

text: T

text: r(Nρˆ

text: (t)).

text: red

text: red

text: Under

text: this

text: circumstance,

text: we

text: can

text: make

text: an

text: eﬀective

text: reduction

text: ρ

text: (t)

text: P

text: (t)

text: ρ

text: (t),

text: r,s;r˜,s˜

text: r,s

text: r,s;r,s

text: obtaining

text: some

text: probability

text: distribution

text: of

text: the

text: collection

text: of

text: states

text: E

text: N

text: This

text: pro-

text: r

text: s

text: vides

text: a

text: “quasi-ensemble”

text: under

text: some

text: (super-operator)

text: evolution

text: resulted

text: from

text: those

text: interactions.

text: Certainly,

text: this

text: “quasi-ensemble”

text: is

text: not

text: the

text: presumed

text: ensemble

text: used

text: in

text: sta-

text: tistical

text: mechanics.

text: The

text: main

text: diﬀerence

text: is

text: that,

text: in

text: the

text: presumed

text: ensemble

text: the

text: member

text: systems

text: are

text: assumed

text: to

text: be

text: uncorrelated

text: or

text: independent

text: While

text: the

text: members

text: in

text: the


text: (cid:80) ∈ M type j. For every machine j , let x be the number the optimal fraction solution of LP to a feasible solution k j ijk (cid:80) of tasks of type i assigned to machine j . Clearly, x = for (4). In [1], the authors obtained a linear program using k ∈ iMj ← ← ∈M x . The ﬁnishing time of a machine j , variable substitution r 1/MS and z x /MS , denko:kted bjy iFjk , is given by k j where MS = max 1 LTB x ETijC isij a lowLBer jk (cid:88) bound on thLeB makespan joMbtjaineid=1byijallowinijg tasks to be divided among all machines. However, the approximation T F = x ETC . (1) of this method would be bad when the objective value is jk ijk ij i=1 close to 0 or little tasks of type i with large ETCij are assigned to machines of type j. A similar phenomenon is Thus, the maximum ﬁnishing time of all machines (i.e., also observed by Tarplee et al. [2]. makespan), denoted by MS(x), is given by To overcome the obstacle mentioned above, we will use a different method. We replace MS(x) with a constant MS, and then obtain an approximate integer linear program (ILP) MS(x) = max max F . (2) j k:jk∈Mj jk for (4). By rounding the optimal fraction solution for the relaxation of ILP based on the classic rounding algorithm for In this paper, for convenience, machines are turned off the generalized assignment problem [3], we obtain a feasible when not use, which means that the energy consumed by solution for (4). It is desired to point out that, in our method, the bag-of-tasks is(cid:88)give(cid:88)n by: the tasks of type i such that ETCij > MS will not be assigned to machines of type j, which is to avoid increasing the makespan too much when rounding the optimal fraction M T E(x) = x APC ETC . (3) solution. ij ij ij j=1 i=1 Let LB be the optimal makespan by ignoring the energy consumption, and UB be the makespan of the feasible Let p be the price customer pays and c be the cost per unit schedule by assigning each task to the machine with min- obfyeelxeecctruitcianlgeanebraggy-.oTf-htaespkrsoiﬁstpth−actEth(exo).rgTahneizEantieorngyre-Acewivaeres icmonusmtanatve(cid:15)rag>e p0o, wCelreacrolyn,sutmheptmioankeAsPpaCnijM. FSo(rxa∗n)yogfivtehne Proﬁt Maximizing Scheduling (EAPMS) Problem deﬁned in optimal solution x∗ for (4) lies in [LB(1+(cid:15))t, LB(1+(cid:15))t+1], [1] attempting to maximize the proﬁt per unit time can for some t ∈ {1, 2, . . . , (cid:100)log UB/LB(cid:101)}. By trying be formulated as the following nonlinear integer program all possible values, we will ﬁn(d1+a(cid:15))feasible makespan MS (NLIP): such that MS(x∗) ∈ [MS/(1 + (cid:15)), MS], where MS = LB(1 + (cid:15))t for some t. For convenience, from now on, − p cE(x) assume that MS is a known constant satisfying Maximize (cid:88) (cid:88) (cid:88) x MS(x) ∗ ≤ ≤ ∗ ∀ M M MS(x ) MS (1 + (cid:15))MS(x ). (5) subject to i x = x = T ; (4) ijk ij i ∈M ∀ j=1 k≤:jk j j=1 ∈ M For a constant MS, as in [1], our algorithm is decomposed j F MS(x), for each j ; jk k j into two phases. This ﬁrst phase rounds the fraction optimal ∀ ∈ Z ∈ M i, j xijk ≥0, for each jk j. solution to obtain a schedule where the numbers x of tasks ij of type i assigned to machines of type j are given. The The objective of (4) is to maximize the proﬁt per unit second phase assigns tasks to actual machines to produce the time, where x is the primary decision variable. The ﬁrst full task allocation x . The next two subsections describe constraint ensures that all tasks of different types in the bag ijk the two phases of this recovery procedure in detail. are assigned to some machine type. Because the objective There are two main differences between Tarplee, Ma- is to maximize the proﬁt per unit time, which is equivalent ciejewski, and Siegel’s (TMS, for short) method [1] and our to minimize makespan, the second constrain ensures that task-type-based (TTB, for short) method (depicted in Figure MS(x) is equal to the maximum ﬁnishing time of all 1.): (1) The TMS method uses one fractional solution to machines. round while we use multiple fractional solutions and choose III. A TASK-TYPE-BASED ALGORITHM the best one; (2) In the ﬁrst phase, the energy consumption A. Overview may increase in Tarplee et al.’s method while it will not Note that (4) is a nonlinear integer program, which can increase in our method. not be solved optimally in polynomial time. To obtain an B. b-Matching-Based Rounding approximate solution of (4), one possible way is to convert Note that if ETC > MS, the tasks of type i can not be ij (4) to an equivalent linear program (LP), and then to round assigned to the machines of type j in the optimal solution, ##LTLine## ##LTLine##

text: M

text: type

text: j.

text: For

text: every

text: machine

text: j

text: let

text: x

text: be

text: the

text: number

text: the

text: optimal

text: fraction

text: solution

text: of

text: LP

text: to

text: a

text: feasible

text: solution

text: k

text: j

text: ijk

text: of

text: tasks

text: of

text: type

text: i

text: assigned

text: to

text: machine

text: j

text: Clearly,

text: x

text: for

text: In

text: the

text: authors

text: obtained

text: a

text: linear

text: program

text: using

text: k

text: iMj

text: ∈M

text: x

text: The

text: ﬁnishing

text: time

text: of

text: a

text: machine

text: j

text: variable

text: substitution

text: r

text: and

text: z

text: x

text: /MS

text: denko:kted

text: bjy

text: iFjk

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text: given

text: by

text: k

text: j

text: where

text: MS

text: max

text: LTB

text: x

text: ETijC

text: isij

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text: lowLBer

text: jk

text: bound

text: on

text: thLeB

text: makespan

text: tasks

text: to

text: be

text: divided

text: among

text: all

text: machines.

text: However,

text: the

text: approximation

text: T

text: F

text: x

text: ETC

text: of

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text: method

text: would

text: be

text: bad

text: when

text: the

text: objective

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text: is

text: jk

text: ijk

text: ij

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text: or

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text: tasks

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text: type

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text: large

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text: are

text: assigned

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text: machines

text: of

text: type

text: j.

text: A

text: similar

text: phenomenon

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text: Thus,

text: the

text: maximum

text: ﬁnishing

text: time

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text: all

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text: (i.e.,

text: also

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text: al.

text: makespan),

text: denoted

text: by

text: MS(x),

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text: To

text: overcome

text: the

text: obstacle

text: mentioned

text: above,

text: we

text: will

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text: a

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text: method.

text: We

text: replace

text: MS(x)

text: with

text: a

text: constant

text: MS,

text: and

text: then

text: obtain

text: an

text: approximate

text: integer

text: linear

text: program

text: (ILP)

text: MS(x)

text: max

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text: F

text: j

text: k:jk∈Mj

text: jk

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text: rounding

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text: optimal

text: fraction

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text: relaxation

text: of

text: ILP

text: based

text: on

text: the

text: classic

text: rounding

text: algorithm

text: for

text: In

text: this

text: paper,

text: for

text: convenience,

text: machines

text: are

text: turned

text: off

text: the

text: generalized

text: assignment

text: problem

text: we

text: obtain

text: a

text: feasible

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text: not

text: use,

text: which

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text: solution

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text: point

text: out

text: that,

text: in

text: our

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text: the

text: bag-of-tasks

text: by:

text: the

text: tasks

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text: type

text: i

text: such

text: that

text: ETCij

text: MS

text: will

text: not

text: be

text: assigned

text: to

text: machines

text: of

text: type

text: j,

text: which

text: is

text: to

text: avoid

text: increasing

text: the

text: makespan

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text: much

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text: rounding

text: the

text: optimal

text: fraction

text: M

text: T

text: E(x)

text: x

text: APC

text: ETC

text: solution.

text: ij

text: ij

text: ij

text: Let

text: LB

text: be

text: the

text: optimal

text: makespan

text: by

text: ignoring

text: the

text: energy

text: consumption,

text: and

text: UB

text: be

text: the

text: makespan

text: of

text: the

text: feasible

text: Let

text: p

text: be

text: the

text: price

text: customer

text: pays

text: and

text: c

text: be

text: the

text: cost

text: per

text: unit

text: schedule

text: by

text: assigning

text: each

text: task

text: to

text: the

text: machine

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text: min-

text: FSo(rxa∗n)yogfivtehne

text: Proﬁt

text: Maximizing

text: Scheduling

text: (EAPMS)

text: Problem

text: deﬁned

text: in

text: optimal

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text: formulated

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text: following

text: nonlinear

text: integer

text: program

text: all

text: possible

text: values,

text: we

text: will

text: makespan

text: MS

text: (NLIP):

text: such

text: that

text: MS(x∗)

text: MS],

text: where

text: MS

text: for

text: some

text: t.

text: For

text: convenience,

text: from

text: now

text: on,

text: p

text: cE(x)

text: assume

text: that

text: MS

text: is

text: a

text: known

text: constant

text: satisfying

text: Maximize

text: x

text: MS(x)

text: M

text: M

text: MS(x

text: MS

text: subject

text: to

text: i

text: x

text: x

text: T

text: ijk

text: ij

text: i

text: ∈M

text: k≤:jk

text: j

text: M

text: For

text: a

text: constant

text: MS,

text: as

text: in

text: our

text: algorithm

text: is

text: decomposed

text: j

text: F

text: MS(x),

text: for

text: each

text: j

text: jk

text: k

text: j

text: into

text: two

text: phases.

text: This

text: ﬁrst

text: phase

text: rounds

text: the

text: fraction

text: optimal

text: Z

text: M

text: i,

text: j

text: xijk

text: for

text: each

text: jk

text: j.

text: solution

text: to

text: obtain

text: a

text: schedule

text: where

text: the

text: numbers

text: x

text: of

text: tasks

text: ij

text: of

text: type

text: i

text: assigned

text: to

text: machines

text: of

text: type

text: j

text: are

text: given.

text: The

text: The

text: objective

text: of

text: is

text: to

text: maximize

text: the

text: proﬁt

text: per

text: unit

text: second

text: phase

text: assigns

text: tasks

text: to

text: actual

text: machines

text: to

text: produce

text: the

text: time,

text: where

text: x

text: is

text: the

text: primary

text: decision

text: variable.

text: The

text: ﬁrst

text: full

text: task

text: allocation

text: x

text: The

text: next

text: two

text: subsections

text: describe

text: constraint

text: ensures

text: that

text: all

text: tasks

text: of

text: different

text: types

text: in

text: the

text: bag

text: ijk

text: the

text: two

text: phases

text: of

text: this

text: recovery

text: procedure

text: in

text: detail.

text: are

text: assigned

text: to

text: some

text: machine

text: type.

text: Because

text: the

text: objective

text: There

text: are

text: two

text: main

text: differences

text: between

text: Tarplee,

text: Ma-

text: is

text: to

text: maximize

text: the

text: proﬁt

text: per

text: unit

text: time,

text: which

text: is

text: equivalent

text: ciejewski,

text: and

text: Siegel’s

text: (TMS,

text: for

text: short)

text: method

text: and

text: our

text: to

text: minimize

text: makespan,

text: the

text: second

text: constrain

text: ensures

text: that

text: task-type-based

text: (TTB,

text: for

text: short)

text: method

text: (depicted

text: in

text: Figure

text: MS(x)

text: is

text: equal

text: to

text: the

text: maximum

text: ﬁnishing

text: time

text: of

text: all

text: The

text: TMS

text: method

text: uses

text: one

text: fractional

text: solution

text: to

text: machines.

text: round

text: while

text: we

text: use

text: multiple

text: fractional

text: solutions

text: and

text: choose

text: III.

text: A

text: TASK-TYPE-BASED

text: ALGORITHM

text: the

text: best

text: one;

text: In

text: the

text: ﬁrst

text: phase,

text: the

text: energy

text: consumption

text: A.

text: Overview

text: may

text: increase

text: in

text: Tarplee

text: et

text: al.’s

text: method

text: while

text: it

text: will

text: not

text: Note

text: that

text: is

text: a

text: nonlinear

text: integer

text: program,

text: which

text: can

text: increase

text: in

text: our

text: method.

text: not

text: be

text: solved

text: optimally

text: in

text: polynomial

text: time.

text: To

text: obtain

text: an

text: B.

text: b-Matching-Based

text: Rounding

text: approximate

text: solution

text: of

text: one

text: possible

text: way

text: is

text: to

text: convert

text: Note

text: that

text: if

text: ETC

text: MS,

text: the

text: tasks

text: of

text: type

text: i

text: can

text: not

text: be

text: ij

text: to

text: an

text: equivalent

text: linear

text: program

text: (LP),

text: and

text: then

text: to

text: round

text: assigned

text: to

text: the

text: machines

text: of

text: type

text: j

text: in

text: the

text: optimal

text: solution,

text: ##LTLine##

text: ##LTLine##


text: Discrete Glimpses of the Physics Landscape after the Higgs Discovery John Ellis Physics Department, King’s College London, Strand, London WC2R 2LS, U.K; Theory Division, Physics Department, CERN, CH 1211 Geneva 23, Switzerland E-mail: John.Ellis@cern.ch Abstract. What is the Higgs boson telling us? What else is there? How do we ﬁnd it? This talk discusses these current topics in particle physics in the wake of the Higgs discovery, with particular emphasis on the discrete symmetries CP and R-parity, not forgetting ﬂavour physics and dark matter, and ﬁnishing with some remarks about possible future colliders. KCL-PH-TH/2015-02, LCTS/2015-01, CERN-PH-TH/2015-008 1. The Story so far Run 1 of the LHC has brought the apotheosis of the Standard Model (SM). The SM has predicted successfully many cross sections for particle and jet production measured at the LHC [1], as seen in Fig. 1. These successes include QCD jet production cross sections, which agree with SM predictions over large ranges in energy and many orders of magnitude, ± measurements of single and multiple W and Z0 production, as well as multiple measurements of top quark production, both pairwise and singly and in association with vector bosons. Moreover, Run 1 of the LHC has also brought the discovery by CMS and ATLAS of a (the?) Higgs boson [2], whose production has by now been observed in three diﬀerent production channels, as also seen in Fig. 1, again with cross sections in agreement with the SM predictions. A major theme of this talk is what we already know about this newly-discovered particle. Meanwhile, over there in ﬂavour space, the Cabibbo-Kobayashi-Maskawa (CKM) description of ﬂavour mixing and CP violation is also a pillar of the SM, but here the picture is more complex. It is in general very successful, as seen in the left panel of Fig. 2 [3], but there are a number of anomalies. On the one hand, the SM predicted successfully the branching ratio for → − the rare decay B µ+µ : s → − × − BR(B µ+µ ) = 2.8+−0.7 10 9 , (1) s 0.6 measured by the CMS and LHCb Collaborations [4]: see Fig. 3. However, the joint CMS and → − LHCb analysis [4] also has an suggestion of a B µ+µ signal that is considerably larger than d the SM prediction (which is an ironclad prediction also of models with minimal ﬂavour violation (MFV), including many SUSY scenarios): → − × − BR(B µ+µ ) = 3.9+−1.6 10 10 , (2) d 1.4

text: Discrete

label: title

text: Glimpses

text: of

text: the

text: Physics

text: Landscape

text: after

text: the

text: Higgs

text: Discovery

text: John

label: author

text: Ellis

text: Physics

text: Department,

text: King’s

text: College

text: London,

text: Strand,

text: London

text: U.K;

text: Theory

text: Division,

text: Physics

text: Department,

text: CERN,

text: CH

text: Geneva

text: Switzerland

text: E-mail:

text: John.Ellis@cern.ch

label: footer

text: Abstract.

text: What

label: abstract

text: is

text: the

text: Higgs

text: boson

text: telling

text: us?

text: What

text: else

text: is

text: there?

text: How

text: do

text: we

text: ﬁnd

text: it?

text: This

text: talk

text: discusses

text: these

text: current

text: topics

text: in

text: particle

text: physics

text: in

text: the

text: wake

text: of

text: the

text: Higgs

text: discovery,

text: with

text: particular

text: emphasis

text: on

text: the

text: discrete

text: symmetries

text: CP

text: and

text: R-parity,

text: not

text: forgetting

text: ﬂavour

text: physics

text: and

text: dark

text: matter,

text: and

text: ﬁnishing

text: with

text: some

text: remarks

text: about

text: possible

text: future

text: colliders.

text: The

text: Story

text: so

text: far

text: Run

text: of

text: the

text: LHC

text: has

text: brought

text: the

text: apotheosis

text: of

text: the

text: Standard

text: Model

text: (SM).

text: The

text: SM

text: has

text: predicted

text: successfully

text: many

text: cross

text: sections

text: for

text: particle

text: and

text: jet

text: production

text: measured

text: at

text: the

text: LHC

text: as

text: seen

text: in

text: Fig.

text: These

text: successes

text: include

text: QCD

text: jet

text: production

text: cross

text: sections,

text: which

text: agree

text: with

text: SM

text: predictions

text: over

text: large

text: ranges

text: in

text: energy

text: and

text: many

text: orders

text: of

text: magnitude,

text: measurements

text: of

text: single

text: and

text: multiple

text: W

text: and

text: production,

text: as

text: well

text: as

text: multiple

text: measurements

text: of

text: top

text: quark

text: production,

text: both

text: pairwise

text: and

text: singly

text: and

text: in

text: association

text: with

text: vector

text: bosons.

text: Moreover,

text: Run

text: of

text: the

text: LHC

text: has

text: also

text: brought

text: the

text: discovery

text: by

text: CMS

text: and

text: ATLAS

text: of

text: a

text: (the?)

text: Higgs

text: boson

text: whose

text: production

text: has

text: by

text: now

text: been

text: observed

text: in

text: three

text: diﬀerent

text: production

text: channels,

text: as

text: also

text: seen

text: in

text: Fig.

text: again

text: with

text: cross

text: sections

text: in

text: agreement

text: with

text: the

text: SM

text: predictions.

text: A

text: major

text: theme

text: of

text: this

text: talk

text: is

text: what

text: we

text: already

text: know

text: about

text: this

text: newly-discovered

text: particle.

text: Meanwhile,

text: over

text: there

text: in

text: ﬂavour

text: space,

text: the

text: Cabibbo-Kobayashi-Maskawa

text: (CKM)

text: description

text: of

text: ﬂavour

text: mixing

text: and

text: CP

text: violation

text: is

text: also

text: a

text: pillar

text: of

text: the

text: SM,

text: but

text: here

text: the

text: picture

text: is

text: more

text: complex.

text: It

text: is

text: in

text: general

text: very

text: successful,

text: as

text: seen

text: in

text: the

text: left

text: panel

text: of

text: Fig.

text: but

text: there

text: are

text: a

text: number

text: of

text: anomalies.

text: On

text: the

text: one

text: hand,

text: the

text: SM

text: predicted

text: successfully

text: the

text: branching

text: ratio

text: for

text: the

text: rare

text: decay

text: B

text: µ+µ

text: s

text: BR(B

text: µ+µ

text: s

text: measured

text: by

text: the

text: CMS

text: and

text: LHCb

text: Collaborations

text: see

text: Fig.

text: However,

text: the

text: joint

text: CMS

text: and

text: LHCb

text: analysis

text: also

text: has

text: an

text: suggestion

text: of

text: a

text: B

text: µ+µ

text: signal

text: that

text: is

text: considerably

text: larger

text: than

text: d

text: the

text: SM

text: prediction

text: (which

text: is

text: an

text: ironclad

text: prediction

text: also

text: of

text: models

text: with

text: minimal

text: ﬂavour

text: violation

text: (MFV),

text: including

text: many

text: SUSY

text: scenarios):

text: BR(B

text: µ+µ

text: d


text: N ve (cid:0) (cid:1) LHC Constraints (cid:0)Operator (cid:1) Coeﬃcient Individual Marginalized O † ↔ = ig H σa DµH DνW a − − − OW 2 (cid:48) † ↔ µν m2W (c c ) ( 0.022, 0.004) ( 0.035, 0.005) = ig H DµH ∂νB Λ2 W B O B = 2ig(DµH)†σa(DνH)Wµνa m2W c (−0.042, 0.008) (−0.035, 0.015) OHW = ig(cid:48)(DµH)†(DνH)B µν mΛ2W2 cHW (−0.053, 0.044) (−0.045, 0.075) OHB µν Λ2 HB − − = 1 g(cid:15) W aνW b W cρµ m2W c ( 0.083, 0.045) ( 0.083, 0.045) 3WO =3!g2|aHbc|2GµA GνAρµν Λm22W c3W (0, 3.0) × 10−5 (−3.2, 1.1) × 10−4 Og = gs(cid:48)2|H|2Bµν Bµν mΛ2W2 cg (−4.0, 2.3) × 10−4 (−11, 2.2) × 10−4 γ µν Λ2 γ Table 2. List of operators entering in LHC Higgs and TGC physics, together with 95% CL bounds when individual coeﬃcients are switched on one at a time, and marginalized in a simultaneous ﬁt [26]. LHC8 ATLAS VH 70 60 50 40 30 20 10 0 0 50 100 (cid:72) 15(cid:76)0 200 250 p GeV T → → Figure 11. Left panel: Simulation of the pV distribution in (V 2(cid:96)) + (H ¯bb) events at T the LHC after implementing ATLAS cuts. The solid distribution is the SM expectation, and the red-dotted and blue-dashed lines correspond to the distributions with c¯ =0.1 and 0.05, W respectively [27]. Right panel: The same-ﬂavour p distribution of the leading lepton after the T TGC analysis cuts for ATLAS at 8 TeV. The Standard Model distribution is shown in blue with solid lines, and the eﬀect of c¯ = 0.1 is superimposed in green with dashed lines. In HW both cases the last (overﬂow) bin provides signiﬁcant extra information compared to the overall normalisation [26]. 4. The SM is not enough! Albert Michelson infamously asserted in 1894 that “The more important fundamental laws and facts of physical science have all been discovered”, just before the discoveries of radioactivity and the electron. Likewise, Lord Kelvin asserted in 1900 that “There is nothing new to be discovered in physics now, all that remains is more and more precise measurement”, just before Einstein postulated the photon and proposed special relativity. Even after the discovery of a (the?) Higgs boson, there are many reasons to expect physics beyond the SM, as I now discuss. Inspired by James Bond [28], here are 007 of them. 1) The measured values of m and m t H suggest that the electroweak vacuum is probably unstable, unless some BSM physics intervenes. 2) The SM cannot provide the dark matter required by astrophysics and cosmology. 3) Additional CP violation beyond the CKM model is required to explain the origin of the matter ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: O

text: ig

text: H

text: σa

text: DµH

text: DνW

text: a

text: OW

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text: DµH

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text: Λ2

text: W

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text: O

text: B

text: c

text: OHW

text: µν

text: cHW

text: OHB

text: µν

text: Λ2

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text: aνW

text: b

text: W

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text: c

text: GνAρµν

text: Og

text: Bµν

text: cg

text: γ

text: µν

text: Λ2

text: γ

text: Table

text: List

text: of

text: operators

text: entering

text: in

text: LHC

text: Higgs

text: and

text: TGC

text: physics,

text: together

text: with

text: CL

text: bounds

text: when

text: individual

text: coeﬃcients

text: are

text: switched

text: on

text: one

text: at

text: a

text: time,

text: and

text: marginalized

text: in

text: a

text: simultaneous

text: ﬁt

text: ATLAS

text: VH

text: p

text: GeV

text: T

text: Figure

text: Left

text: panel:

text: Simulation

text: of

text: the

text: pV

text: distribution

text: in

text: (V

text: (H

text: ¯bb)

text: events

text: at

text: T

text: the

text: LHC

text: after

text: implementing

text: ATLAS

text: cuts.

text: The

text: solid

text: distribution

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text: the

text: SM

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text: and

text: the

text: red-dotted

text: and

text: blue-dashed

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text: correspond

text: to

text: the

text: distributions

text: with

text: c¯

text: and

text: W

text: respectively

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text: panel:

text: The

text: same-ﬂavour

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text: distribution

text: of

text: the

text: leading

text: lepton

text: after

text: the

text: T

text: TGC

text: analysis

text: cuts

text: for

text: ATLAS

text: at

text: TeV.

text: The

text: Standard

text: Model

text: distribution

text: is

text: shown

text: in

text: blue

text: with

text: solid

text: lines,

text: and

text: the

text: eﬀect

text: of

text: c¯

text: is

text: superimposed

text: in

text: green

text: with

text: dashed

text: lines.

text: In

text: HW

text: both

text: cases

text: the

text: last

text: (overﬂow)

text: bin

text: provides

text: signiﬁcant

text: extra

text: information

text: compared

text: to

text: the

text: overall

text: normalisation

text: The

text: SM

text: is

text: not

text: enough!

text: Albert

text: Michelson

text: infamously

text: asserted

text: in

text: that

text: “The

text: more

text: important

text: fundamental

text: laws

text: and

text: facts

text: of

text: physical

text: science

text: have

text: all

text: been

text: discovered”,

text: just

text: before

text: the

text: discoveries

text: of

text: radioactivity

text: and

text: the

text: electron.

text: Likewise,

text: Lord

text: Kelvin

text: asserted

text: in

text: that

text: “There

text: is

text: nothing

text: new

text: to

text: be

text: discovered

text: in

text: physics

text: now,

text: all

text: that

text: remains

text: is

text: more

text: and

text: more

text: precise

text: measurement”,

text: just

text: before

text: Einstein

text: postulated

text: the

text: photon

text: and

text: proposed

text: special

text: relativity.

text: Even

text: after

text: the

text: discovery

text: of

text: a

text: (the?)

text: Higgs

text: boson,

text: there

text: are

text: many

text: reasons

text: to

text: expect

text: physics

text: beyond

text: the

text: SM,

text: as

text: I

text: now

text: discuss.

text: Inspired

text: by

text: James

text: Bond

text: here

text: are

text: of

text: them.

text: The

text: measured

text: values

text: of

text: m

text: and

text: m

text: t

text: H

text: suggest

text: that

text: the

text: electroweak

text: vacuum

text: is

text: probably

text: unstable,

text: unless

text: some

text: BSM

text: physics

text: intervenes.

text: The

text: SM

text: cannot

text: provide

text: the

text: dark

text: matter

text: required

text: by

text: astrophysics

text: and

text: cosmology.

text: Additional

text: CP

text: violation

text: beyond

text: the

text: CKM

text: model

text: is

text: required

text: to

text: explain

text: the

text: origin

text: of

text: the

text: matter

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text: 2 (cid:90) (cid:20) (cid:21) √ A − R − ∂ φ∂µφ 16π L = d4x g φ [g] ω µ + , (2.1) φ c4 m R L where [g] is the scalar curvature function of the metric g , and is the ordinary matter contribution. The µν m constant ω is determined by observations and its value can be constrained by classical tests of General Relativity. The consequences of BD action (2.1) have been analyzed for the light deﬂection, the relativistic perihelion rotation of Mercury, and the time delay experiments, resulting in reasonable agreement with all available observations thus ≥ far provided (ω 500)[21]. The most recent bound ω > 3000 has been derived by Very-Long Baseline Radio Interferometry (VLBI) experiment [22]. On the other hand, bounds on the anisotropy of the microwave background ≤ radiation give the upper limit ω 301[7]. Nevertheless, as shown in Refs. [25, 26], the consequences of the BD theory in the radiation-matter equality, seem to indicate that at the moment there is no observable eﬀects appearing from → ∞ CMBR. Einstein’s theory is fully recovered for ω . In this limit, the BD theory becomes indistinguishable from General Relativity in all its predictions. The variation of the action (2.1) with respect to the metric tensor g and µν the scalar ﬁeld φ yields the ﬁeld equations [1] (cid:18) (cid:19) R − 1 R 8π ω − 1 1 − (cid:3) g = T + φ φ g φ φ,α + (φ g φ) (2.2) µν 2 µν c4φ µν φ2 ,µ ,ν 2 µν ,α φ ,µ;ν µν and 2ω (cid:3) − ω R φ φ φ,µ + = 0 . (2.3) φ φ2 ,µ (cid:3) Here is the d’Alembert operator in curved spacetime. T is the energy-momentum tensor of matter. The line element describing a static and isotropic geometry is µν − ds2 = eνdt2 + eµ[dr2 + r2(dθ2 + sin2 θdϕ2)] , (2.4) where the functions ν and µ depend on the radial coordinate r. For r > B, the general solution in the vacuum is given by (cid:20) (cid:21) − (cid:18)1 B/(cid:19)r (cid:20)2/λ (cid:21) eν = e2α0 , (2.5) 1 + B/r − − − (cid:20) B (cid:21) 4 1 B/r 2(λ C 1)/λ eµ = e2β0 1 + , (2.6) r 1 + B/r − − 1 B/r C/λ φ = φ , (2.7) 0 1 + B/r where α0, β0, φ0, B, and C are arbitrary c(cid:20)onstants. B is(cid:18)related to(cid:19)t(cid:21)he mass of the source. Here ω > 3/2 and − − ωC 1/2 λ = (C + 1)2 C 1 . (2.8) 2 In the weak ﬁeld approximation, the constraints on BD parameters are [1] 1 This dichotomy in the bounds of ω of the original BD scheme has led to take into account self-interaction scalar ﬁeld potential (see e.g. [23, 24].) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: R

text: φ∂µφ

text: L

text: g

text: φ

text: [g]

text: ω

text: µ

text: φ

text: m

text: R

text: L

text: where

text: [g]

text: is

text: the

text: scalar

text: curvature

text: function

text: of

text: the

text: metric

text: g

text: and

text: is

text: the

text: ordinary

text: matter

text: contribution.

text: The

text: µν

text: m

text: constant

text: ω

text: is

text: determined

text: by

text: observations

text: and

text: its

text: value

text: can

text: be

text: constrained

text: by

text: classical

text: tests

text: of

text: General

text: Relativity.

text: The

text: consequences

text: of

text: BD

text: action

text: have

text: been

text: analyzed

text: for

text: the

text: light

text: deﬂection,

text: the

text: relativistic

text: perihelion

text: rotation

text: of

text: Mercury,

text: and

text: the

text: time

text: delay

text: experiments,

text: resulting

text: in

text: reasonable

text: agreement

text: with

text: all

text: available

text: observations

text: thus

text: far

text: provided

text: (ω

text: The

text: most

text: recent

text: bound

text: ω

text: has

text: been

text: derived

text: by

text: Very-Long

text: Baseline

text: Radio

text: Interferometry

text: (VLBI)

text: experiment

text: On

text: the

text: other

text: hand,

text: bounds

text: on

text: the

text: anisotropy

text: of

text: the

text: microwave

text: background

text: radiation

text: give

text: the

text: upper

text: limit

text: ω

text: Nevertheless,

text: as

text: shown

text: in

text: Refs.

text: the

text: consequences

text: of

text: the

text: BD

text: theory

text: in

text: the

text: radiation-matter

text: equality,

text: seem

text: to

text: indicate

text: that

text: at

text: the

text: moment

text: there

text: is

text: no

text: observable

text: eﬀects

text: appearing

text: from

text: CMBR.

text: Einstein’s

text: theory

text: is

text: fully

text: recovered

text: for

text: ω

text: In

text: this

text: limit,

text: the

text: BD

text: theory

text: becomes

text: indistinguishable

text: from

text: General

text: Relativity

text: in

text: all

text: its

text: predictions.

text: The

text: variation

text: of

text: the

text: action

text: with

text: respect

text: to

text: the

text: metric

text: tensor

text: g

text: and

text: µν

text: the

text: scalar

text: ﬁeld

text: φ

text: yields

text: the

text: ﬁeld

text: equations

text: R

text: R

text: 8π

text: ω

text: g

text: T

text: φ

text: φ

text: g

text: φ

text: φ,α

text: (φ

text: g

text: φ)

text: µν

text: µν

text: µν

text: φ2

text: ,µ

text: ,ν

text: µν

text: ,α

text: φ

text: ,µ;ν

text: µν

text: and

text: 2ω

text: ω

text: R

text: φ

text: φ

text: φ,µ

text: φ

text: φ2

text: ,µ

text: Here

text: is

text: the

text: d’Alembert

text: operator

text: in

text: curved

text: spacetime.

text: T

text: is

text: the

text: energy-momentum

text: tensor

text: of

text: matter.

text: The

text: line

text: element

text: describing

text: a

text: static

text: and

text: isotropic

text: geometry

text: is

text: µν

text: where

text: the

text: functions

text: ν

text: and

text: µ

text: depend

text: on

text: the

text: radial

text: coordinate

text: r.

text: For

text: r

text: B,

text: the

text: general

text: solution

text: in

text: the

text: vacuum

text: is

text: given

text: by

text: eν

text: B/r

text: B

text: B/r

text: C

text: eµ

text: r

text: B/r

text: B/r

text: C/λ

text: φ

text: φ

text: B/r

text: where

text: B,

text: and

text: C

text: are

text: arbitrary

text: B

text: mass

text: of

text: the

text: source.

text: Here

text: ω

text: and

text: ωC

text: λ

text: (C

text: C

text: In

text: the

text: weak

text: ﬁeld

text: approximation,

text: the

text: constraints

text: on

text: BD

text: parameters

text: are

text: This

text: dichotomy

text: in

text: the

text: bounds

text: of

text: ω

text: of

text: the

text: original

text: BD

text: scheme

text: has

text: led

text: to

text: take

text: into

text: account

text: self-interaction

text: scalar

text: ﬁeld

text: potential

text: (see

text: e.g.

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text: 9 (cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) | | ≈ B2 F x 2(2+C)/λ 1 ≈ × − F GeV B 2 x km 2(2+C)/λ km 5 a 1.2 10 48 eV , (5.9) 8m2 2B x5 m km km 2B x 0 0 (cid:18) (cid:19) (cid:18) (cid:19) where F ≡ C − ωC − − C 1 λ 1 . λ3 2 2 The emitted energy by the charged particl(cid:18)e is therefo(cid:19)re(cid:18) (cid:19) (cid:18) (cid:19) ≈ − F GeV B 4 x km 4(2+C)/λ km 10 W 10 97 2 eV2 , (5.10) m km km 2B x 0 depending on the parameters of the BD spherically symmetric solution. The mechanism for generating photons with high energy is strictly related to the (inﬁnite) potential well near the singularity r = B. The latter occurs owing to the presence of the scalar ﬁeld as source of the gravitational ﬁeld. The issue is: Do naked singularities exist or form during the gravitational collapse? According to the cosmic censorship hypothesis, the formation of naked singularities is forbidden [31]. Nevertheless, two comments are in order at this point: i) the validity of this conjecture is still an open issue in General Relativity (there exist many counterexamples that violate such a conjecture, see, for example, [32, 33]); ii) it is expected that a full quantum theory of gravity may prevent the occurrence of singularity. At the ∼ − Planck scale, characterized by the scale length l 10 33cm, the quantum (gravitational) eﬀects become important, P since they allow to solve the short-distance problems of General Relativity by smearing out the classical singularities. Thus, at this distance, one has a ﬁnite potential well, and the emitted energy is the maximum one. (cid:39) By requiring that the maximum emitted energy is comparable with the observed GRBs energy, i.e. W W , | | (cid:46) GRBs one can ﬁx the parameter C (actually, we are determining an upper bound on C , since W W ). Here ∼ (cid:39) ∼ ≈ GRBs W 1054 erg/s 1051 eV2. Estimations are carried out for protons, m 1 GeV, and B 3 km (the mass GRBs 0 source is of the order of the solar mass). Taking into account all these features and ﬁxing ω = 5000, we ﬁnd that the (cid:39) ∼ parameter C turns out to be of the order C 0.01815. In the case where ω 500, the parameter C assumes the ∼ value C 0.055. Therefore a small deviation from General Relativity (where C = 0) allows the production of the (cid:18) (cid:19) (cid:18) (cid:19) huge amount of energy, typical of GRBs. For the sake of completeness, let us evaluate the energy emission of charged particles near the event horizon, i.e. in (cid:18) (cid:19) |(cid:18)| (cid:19)(cid:39) 1 ∼ − GeV 2 km 3 the framework of General Relativity. The acceleration is a 10 51 eV, so that the GR 128m2B3 m B ∼ − GeV 4 km 6 emitted energy turns out to be W 10 102 , which is negligible in comparison to the observed GR m B energies of GRBs. Finally, a comment is in order. The isotropy and homogeneity of matter distribution in the Universe suggest that gravitational collapse or the emission of radiation by highly accelerated charged particles generate bursts of photons that are isotropically distributed. Thus, alternative theories of gravity seem to be a suitable framework for future investigations of GRBs physics. 6. CONCLUSIONS In this paper we have investigated the propagation of quantum particles in the framework of a spherically symmetric solutions achieved in the framework of BD theory of gravity. Such a propagation could explain some features of GRBs physics. Speciﬁcally, we have shown that in the BD geometry, (scalar) particles near the naked singularity suﬀer high accelerations, producing photons with an extraordinarily large energy output (emission from naked singularity, the latter being regularized by quantum ﬂuctuations occurring at the Planck scale). This is in agreement with the fact that GRBs spectrum is nonthermal, and, as widely believed, it is probably due to the synchrotron radiation. Conversely, GRBs could provide a direct method for testing strong gravity, which, in turn, might provide a formidable tool in order to distinguish alternative theories of gravity from General Relativity. Nevertheless, a lot of work is still necessary for a complete understanding of the rough model here proposed. One of the basic issues is certainly represented by a careful analysis of the back-reaction eﬀects, including the radiation by BD scalar ﬁeld, which could play a nontrivial role on the structure of the mathematical formalism used here. Finally, we want to stress again that the BD theory is only a prototype of large families of models including scalar- tensor gravity with self-interaction potentials, f(R) gravity, Gauss-Bonnet gravity [34], hybrid gravity [24] and so on ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F

text: x

text: F

text: GeV

text: B

text: x

text: km

text: km

text: a

text: eV

text: m

text: km

text: km

text: x

text: where

text: F

text: C

text: ωC

text: C

text: λ

text: λ3

text: The

text: emitted

text: energy

text: by

text: the

text: charged

text: is

text: F

text: GeV

text: B

text: x

text: km

text: km

text: W

text: m

text: km

text: km

text: x

text: depending

text: on

text: the

text: parameters

text: of

text: the

text: BD

text: spherically

text: symmetric

text: solution.

text: The

text: mechanism

text: for

text: generating

text: photons

text: with

text: high

text: energy

text: is

text: strictly

text: related

text: to

text: the

text: (inﬁnite)

text: potential

text: well

text: near

text: the

text: singularity

text: r

text: B.

text: The

text: latter

text: occurs

text: owing

text: to

text: the

text: presence

text: of

text: the

text: scalar

text: ﬁeld

text: as

text: source

text: of

text: the

text: gravitational

text: ﬁeld.

text: The

text: issue

text: is:

text: Do

text: naked

text: singularities

text: exist

text: or

text: form

text: during

text: the

text: gravitational

text: collapse?

text: According

text: to

text: the

text: cosmic

text: censorship

text: hypothesis,

text: the

text: formation

text: of

text: naked

text: singularities

text: is

text: forbidden

text: Nevertheless,

text: two

text: comments

text: are

text: in

text: order

text: at

text: this

text: point:

text: i)

text: the

text: validity

text: of

text: this

text: conjecture

text: is

text: still

text: an

text: open

text: issue

text: in

text: General

text: Relativity

text: (there

text: exist

text: many

text: counterexamples

text: that

text: violate

text: such

text: a

text: conjecture,

text: see,

text: for

text: example,

text: ii)

text: it

text: is

text: expected

text: that

text: a

text: full

text: quantum

text: theory

text: of

text: gravity

text: may

text: prevent

text: the

text: occurrence

text: of

text: singularity.

text: At

text: the

text: Planck

text: scale,

text: characterized

text: by

text: the

text: scale

text: length

text: l

text: the

text: quantum

text: (gravitational)

text: eﬀects

text: become

text: important,

text: P

text: since

text: they

text: allow

text: to

text: solve

text: the

text: short-distance

text: problems

text: of

text: General

text: Relativity

text: by

text: smearing

text: out

text: the

text: classical

text: singularities.

text: Thus,

text: at

text: this

text: distance,

text: one

text: has

text: a

text: ﬁnite

text: potential

text: well,

text: and

text: the

text: emitted

text: energy

text: is

text: the

text: maximum

text: one.

text: By

text: requiring

text: that

text: the

text: maximum

text: emitted

text: energy

text: is

text: comparable

text: with

text: the

text: observed

text: GRBs

text: energy,

text: i.e.

text: W

text: W

text: GRBs

text: one

text: can

text: ﬁx

text: the

text: parameter

text: C

text: (actually,

text: we

text: are

text: determining

text: an

text: upper

text: bound

text: on

text: C

text: since

text: W

text: W

text: Here

text: GRBs

text: W

text: erg/s

text: Estimations

text: are

text: carried

text: out

text: for

text: protons,

text: m

text: GeV,

text: and

text: B

text: km

text: (the

text: mass

text: GRBs

text: source

text: is

text: of

text: the

text: order

text: of

text: the

text: solar

text: mass).

text: Taking

text: into

text: account

text: all

text: these

text: features

text: and

text: ﬁxing

text: ω

text: we

text: ﬁnd

text: that

text: the

text: parameter

text: C

text: turns

text: out

text: to

text: be

text: of

text: the

text: order

text: C

text: In

text: the

text: case

text: where

text: ω

text: the

text: parameter

text: C

text: assumes

text: the

text: value

text: C

text: Therefore

text: a

text: small

text: deviation

text: from

text: General

text: Relativity

text: (where

text: C

text: allows

text: the

text: production

text: of

text: the

text: huge

text: amount

text: of

text: energy,

text: typical

text: of

text: GRBs.

text: For

text: the

text: sake

text: of

text: completeness,

text: let

text: us

text: evaluate

text: the

text: energy

text: emission

text: of

text: charged

text: particles

text: near

text: the

text: event

text: horizon,

text: i.e.

text: in

text: GeV

text: km

text: the

text: framework

text: of

text: General

text: Relativity.

text: The

text: acceleration

text: is

text: a

text: eV,

text: so

text: that

text: the

text: GR

text: m

text: B

text: GeV

text: km

text: emitted

text: energy

text: turns

text: out

text: to

text: be

text: W

text: which

text: is

text: negligible

text: in

text: comparison

text: to

text: the

text: observed

text: GR

text: m

text: B

text: energies

text: of

text: GRBs.

text: Finally,

text: a

text: comment

text: is

text: in

text: order.

text: The

text: isotropy

text: and

text: homogeneity

text: of

text: matter

text: distribution

text: in

text: the

text: Universe

text: suggest

text: that

text: gravitational

text: collapse

text: or

text: the

text: emission

text: of

text: radiation

text: by

text: highly

text: accelerated

text: charged

text: particles

text: generate

text: bursts

text: of

text: photons

text: that

text: are

text: isotropically

text: distributed.

text: Thus,

text: alternative

text: theories

text: of

text: gravity

text: seem

text: to

text: be

text: a

text: suitable

text: framework

text: for

text: future

text: investigations

text: of

text: GRBs

text: physics.

text: CONCLUSIONS

text: In

text: this

text: paper

text: we

text: have

text: investigated

text: the

text: propagation

text: of

text: quantum

text: particles

text: in

text: the

text: framework

text: of

text: a

text: spherically

text: symmetric

text: solutions

text: achieved

text: in

text: the

text: framework

text: of

text: BD

text: theory

text: of

text: gravity.

text: Such

text: a

text: propagation

text: could

text: explain

text: some

text: features

text: of

text: GRBs

text: physics.

text: Speciﬁcally,

text: we

text: have

text: shown

text: that

text: in

text: the

text: BD

text: geometry,

text: (scalar)

text: particles

text: near

text: the

text: naked

text: singularity

text: suﬀer

text: high

text: accelerations,

text: producing

text: photons

text: with

text: an

text: extraordinarily

text: large

text: energy

text: output

text: (emission

text: from

text: naked

text: singularity,

text: the

text: latter

text: being

text: regularized

text: by

text: quantum

text: ﬂuctuations

text: occurring

text: at

text: the

text: Planck

text: scale).

text: This

text: is

text: in

text: agreement

text: with

text: the

text: fact

text: that

text: GRBs

text: spectrum

text: is

text: nonthermal,

text: and,

text: as

text: widely

text: believed,

text: it

text: is

text: probably

text: due

text: to

text: the

text: synchrotron

text: radiation.

text: Conversely,

text: GRBs

text: could

text: provide

text: a

text: direct

text: method

text: for

text: testing

text: strong

text: gravity,

text: which,

text: in

text: turn,

text: might

text: provide

text: a

text: formidable

text: tool

text: in

text: order

text: to

text: distinguish

text: alternative

text: theories

text: of

text: gravity

text: from

text: General

text: Relativity.

text: Nevertheless,

text: a

text: lot

text: of

text: work

text: is

text: still

text: necessary

text: for

text: a

text: complete

text: understanding

text: of

text: the

text: rough

text: model

text: here

text: proposed.

text: One

text: of

text: the

text: basic

text: issues

text: is

text: certainly

text: represented

text: by

text: a

text: careful

text: analysis

text: of

text: the

text: back-reaction

text: eﬀects,

text: including

text: the

text: radiation

text: by

text: BD

text: scalar

text: ﬁeld,

text: which

text: could

text: play

text: a

text: nontrivial

text: role

text: on

text: the

text: structure

text: of

text: the

text: mathematical

text: formalism

text: used

text: here.

text: Finally,

text: we

text: want

text: to

text: stress

text: again

text: that

text: the

text: BD

text: theory

text: is

text: only

text: a

text: prototype

text: of

text: large

text: families

text: of

text: models

text: including

text: scalar-

text: tensor

text: gravity

text: with

text: self-interaction

text: potentials,

text: f(R)

text: gravity,

text: Gauss-Bonnet

text: gravity

text: hybrid

text: gravity

text: and

text: so

text: on

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