text: 10 In view of this fact, we can apply the general approximation-based method in [38] to analyze the performance of the plug-in approach. By construction it is obvious that g (q) is continuously differentiable on [0, 1], which coincides with g(q) = ln q on [ 1 , 1]. n n Moreover, since qˆ is a multiple of 1 , g (qˆ ) only differs from ln( 1 ∨qˆ ) at qˆ = 0 by |−(1+ln n)−ln(1/n)| = 1. Hence, we i n n i n i i may consider the performance of the plug-in estimator pˆ (ln pˆ − g (qˆ )) in estimating p(ln p − g (q)), which is summarized i i n i n in the following lemma. Lemma 1. Let npˆ ∼ Poi(mp) and nqˆ ∼ Poi(nq) be independent, and p ≤ u(S)q. Then we have 30u(S) 5 ln 2 |E[pˆ(ln pˆ− g (qˆ))] − p(ln p − g (q))| ≤ + , (37) n n n m (cid:18) (cid:19) (cid:18) (cid:19) 51 2 4q 4u(S) 700u(S) 1 Var(pˆ(ln pˆ− g (qˆ)) ≤ + p + p(ln u(S))2 + + + p + . (38) n m2 m e2 en n m In particular, u(S) 1 |E[pˆ(ln pˆ− g (qˆ))] − p(ln p − g (q))| (cid:46) + , (39) n n n m 1 pu(S) u(S) p(1 + ln u(S))2 q Var(pˆ(ln pˆ− g (qˆ))) (cid:46) + + + + . (40) n m2 n mn m m Hence, by Lemma 1, we conclude that (cid:88)S |ED (P (cid:107)Q ) − D (P(cid:107)Q)| ≤ |E[pˆ (ln pˆ − g (qˆ))] − p (ln p − g (q ))| (41) 1 m n 1 i i n i i i n i i=1 (cid:88)S u(S) 1 (cid:46) + (42) n m i=1 Su(S) S = + (43) n m and (cid:88)S Var(D (P (cid:107)Q )) = Var(pˆ g (qˆ )) (44) 1 m n i n i i=1 (cid:88)S 1 p u(S) u(S) p (1 + ln u(S))2 q (cid:46) + i + + i + i (45) m2 n mn m m i=1 S u(S) Su(S) (1 + ln u(S))2 (cid:46) + + + (46) m2 n mn m (cid:18) (cid:19) S u(S) (Su(S))2 1 (ln u(S))2 (cid:46) + + + + (47) m2 n n2 m2 m S u(S) (Su(S))2 (ln u(S))2 (cid:46) + + + . (48) m2 n n2 m Combining these two inequalities yields, for any (P, Q) ∈ U , S,u(S) E (D (P (cid:107)Q ) − D (P(cid:107)Q))2 = |ED (P (cid:107)Q ) − D (P(cid:107)Q)|2 + Var(D (P (cid:107)Q )) (49) 1 m n 1 1 m n 1 1 m n (cid:18) (cid:19) Su(S) S 2 u(S) (ln u(S))2 (cid:46) + + + . (50) n m n m To prove Theorem 2, it remains to compute the difference between D and D . By the deﬁnition of g (·), we have 1 n ##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##

text: In

text: view

text: of

text: this

text: fact,

text: we

text: can

text: apply

text: the

text: general

text: approximation-based

text: method

text: in

text: to

text: analyze

text: the

text: performance

text: of

text: the

text: plug-in

text: approach.

text: By

text: construction

text: it

text: is

text: obvious

text: that

text: g

text: (q)

text: is

text: continuously

text: differentiable

text: on

text: which

text: coincides

text: with

text: g(q)

text: ln

text: q

text: on

text: n

text: n

text: Moreover,

text: since

text: qˆ

text: is

text: a

text: multiple

text: of

text: g

text: (qˆ

text: only

text: differs

text: from

text: ln(

text: ∨qˆ

text: at

text: qˆ

text: by

text: Hence,

text: we

text: i

text: n

text: n

text: i

text: n

text: i

text: i

text: may

text: consider

text: the

text: performance

text: of

text: the

text: plug-in

text: estimator

text: pˆ

text: (ln

text: pˆ

text: g

text: (qˆ

text: in

text: estimating

text: p(ln

text: p

text: g

text: (q)),

text: which

text: is

text: summarized

text: i

text: i

text: n

text: i

text: n

text: in

text: the

text: following

text: lemma.

text: Lemma

text: Let

text: npˆ

text: Poi(mp)

text: and

text: nqˆ

text: Poi(nq)

text: be

text: independent,

text: and

text: p

text: u(S)q.

text: Then

text: we

text: have

text: ln

text: |E[pˆ(ln

text: pˆ−

text: g

text: (qˆ))]

text: p(ln

text: p

text: g

text: (q))|

text: n

text: n

text: n

text: m

text: Var(pˆ(ln

text: pˆ−

text: g

text: (qˆ))

text: p

text: p(ln

text: p

text: n

text: m

text: en

text: n

text: m

text: In

text: particular,

text: u(S)

text: |E[pˆ(ln

text: pˆ−

text: g

text: (qˆ))]

text: p(ln

text: p

text: g

text: (q))|

text: n

text: n

text: n

text: m

text: pu(S)

text: u(S)

text: ln

text: q

text: Var(pˆ(ln

text: pˆ−

text: g

text: (qˆ)))

text: n

text: n

text: mn

text: m

text: m

text: Hence,

text: by

text: Lemma

text: we

text: conclude

text: that

text: |ED

text: (P

text: D

text: |E[pˆ

text: (ln

text: pˆ

text: g

text: (qˆ))]

text: p

text: (ln

text: p

text: g

text: (q

text: m

text: n

text: i

text: i

text: n

text: i

text: i

text: i

text: n

text: i

text: u(S)

text: n

text: m

text: Su(S)

text: S

text: n

text: m

text: and

text: Var(D

text: (P

text: Var(pˆ

text: g

text: (qˆ

text: m

text: n

text: i

text: n

text: i

text: p

text: u(S)

text: u(S)

text: p

text: ln

text: q

text: i

text: i

text: i

text: n

text: mn

text: m

text: m

text: S

text: u(S)

text: Su(S)

text: ln

text: n

text: mn

text: m

text: S

text: u(S)

text: (ln

text: n

text: m

text: S

text: u(S)

text: (ln

text: n

text: m

text: Combining

text: these

text: two

text: inequalities

text: yields,

text: for

text: any

text: (P,

text: Q)

text: U

text: S,u(S)

text: E

text: (D

text: (P

text: D

text: |ED

text: (P

text: D

text: Var(D

text: (P

text: m

text: n

text: m

text: n

text: m

text: n

text: Su(S)

text: S

text: u(S)

text: (ln

text: n

text: m

text: n

text: m

text: To

text: prove

text: Theorem

text: it

text: remains

text: to

text: compute

text: the

text: difference

text: between

text: D

text: and

text: D

text: By

text: the

text: deﬁnition

text: of

text: g

text: we

text: have

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: 4 (cid:82)(cid:82) for large r. The scattering states for a linear potential H is Hermitian with respect to the scalar product oscillate faster for large distance, such that they cannot (cid:104) eﬀ | (cid:105) ∗ f(r, θ) g(r, θ) = rdr dθ f (r, θ)g(r, θ). be accurately described within a discretization scheme. B. Numerical implementation On the other hand, the overlap with the bound states at very large distance is negligible, so that the exact be- The Hamiltonian H is invariant under spatial ro- havior barely impacts the position and the width of the eﬀ tations, therefore the eigenstates can be written as resonance. Therefore to avoid numerical artifacts and eimθ(ψ (r), ψ (r), ψ−(r)). For each value of m, the re- increase the numerical accuracy, we replace the antitrap- sulting+Hamilt0onian H only depends on the radial coor- ping potential for ψ− by (cid:114) m dinate r. Since the value of the functions ψ (r) at r = 0 j is not ﬁxed by any boundary condition, one uses a dis- 1 r2 cretization scheme that does not contain the point r = 0, V (r) = 1 + ρ2 , (26) scatt ρ2 1 + h2r2 namely the grid points are r = (n+1/2)∆r, for n rang- n ing from 0 to a maximum value N. 1/∆r ﬁxes the number ≈ − where h is a small parameter, typically h 10 2. For of grid points per harmonic length, whereas N∆r corre- (cid:28) ≈ (cid:29) (cid:82) (cid:0) (cid:1) hr 1, then V (r) V (r), whereas for hr 1, then sponds to the size of the system in harmonic length units. ≈ scatt V (r) 1/hρ, such that the scattering states have The Hamiltonian H is Hermitian for the scalar product scatt (cid:104) | (cid:105) ∗ m ∗ ∗ a well deﬁned wavelength for large r values. We have F G = rdr f (r)g (r) + f (r)g (r) + f−(r)g−(r) , + + 0 0 numerically checked that the results presented here are where F = (f (r), f (r), f−(r)), such that it is the dis- + 0 insensitive to the actual h value. cretization of the equation rH ψ = Erψ that leads to m a generalized eigenvalue problem AX = EBX, where A × and B are (3N + 3) (3N + 3) Hermitian matrices, B C. Results being positive deﬁnite. Neglecting the oﬀ-diagonal coupling, the eﬀective po- tentials seen by ψ and ψ− components being 0 and The properties of the resonances (position in energy − 0 and width) are displayed, as functions of ω /ω , by the V (r), the corresponding eigenstates are not bound T L Fig. 1 for the m = 0 states and by the Fig. 2 for the states but scattering states, whereas they are bound ± m = 1 states. In both cases, the zero of energy cor- states for the ψ component. The oﬀ-diagonal coupling + responds to the bottom of the trapping state, i.e., cor- allows this bound states to decay to the scattering chan- − responding to a global shift of 1 of the eigenergies of nels, which, from a mathematical point of view, becomes ρ2 → resonances, i.e−., complex poles of the Green function Heﬀ, such that in the adiabatic limit ρ 0, the ener- G(z) = 1/(z H). The complex rotation method is gies directly correspond to the harmonic oscillator levels. appropriate to compute directly the properties of these This is clearly seen≈on the left part of the top plots. On resonances (energy, width). Its properties rely on math- the contrary, for ρ 1 (right part), the energy levels dif- ematical properties of the analytic continuation of the fer from the ha−rmonic one, in particular, the diﬀerence Green’s function in the complex plane [14, 15]. A review in energy En+1 En gets smaller with higher n reﬂecting of its application to atomic physics can be found in [10]. the linear behavior of the trapping potential V (r) at large The method is implemented in our case by making the r. The behavior in the adiabatic regime can be obtained → → − radial coordinate complex: r eiφr and ∂ e iφ ∂ , from the perturbation expansion of h++ with respect to where φ is a real parameter (the rotation∂rangle). T∂hre A+ and φ+. More precisely, for a ﬁxed value of m, h++ matrix representations of the Hamiltonian then become reads: (cid:18) (cid:19) (cid:18) (cid:19) complex symmetric, but are no more Hermitian. The − − fund•amThenetbaol upnrodpsetrattieess,oiff tahney,caormepslteixll sopnectthrearaerael :axis. h++ = h(+0+) −rm2 1 − γ1 +41r2 3γ4 4γ3γ4 2γ2 + 1 . • (27) The continua are rotated by an angle 2φ on the lower-half complex plane, around their branching The second term arises from the gauge ﬁeld A and point. ± + results in a energy splitting among the m states, which • Each complex eigenvalue Ej gives the properties of is clearly observed in Fig. 2. A Ta(cid:28)ylor expansion of these one resonance, i.e., the energy is the real part of two terms at small distances ρr 1 leads respectively E , and the width is two times the negative of the to: (cid:18) (cid:19) (cid:18) (cid:19) j imaginary part. The complex eigenvalues are inde- pendent of φ, provided that they are not covered (cid:18) −m − 1 (cid:19) ≈ − ρ2 − 3 by the continua. 1 m ρ4r2 r2 γ 2 8 − − Note that in the present case, the branching point asso- 1 3γ4 4γ3 2γ2 + 1 ≈ ρ2 − 5 ρ4r2 ciated to ψ is located at E = 0, whereas the one associ- 4r2 γ4 2 8 0 −∞ − ≈ − ated to ψ− is formally at E = , since V (r) r/ρ (28) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: for

text: large

text: r.

text: The

text: scattering

text: states

text: for

text: a

text: linear

text: potential

text: H

text: is

text: Hermitian

text: with

text: respect

text: to

text: the

text: scalar

text: product

text: oscillate

text: faster

text: for

text: large

text: distance,

text: such

text: that

text: they

text: cannot

text: eﬀ

text: f(r,

text: θ)

text: g(r,

text: θ)

text: rdr

text: dθ

text: f

text: (r,

text: θ)g(r,

text: θ).

text: be

text: accurately

text: described

text: within

text: a

text: discretization

text: scheme.

text: B.

label: section

text: Numerical

text: implementation

text: On

text: the

text: other

text: hand,

text: the

text: overlap

text: with

text: the

text: bound

text: states

text: at

text: very

text: large

text: distance

text: is

text: negligible,

text: so

text: that

text: the

text: exact

text: be-

text: The

text: Hamiltonian

text: H

text: is

text: invariant

text: under

text: spatial

text: ro-

text: havior

text: barely

text: impacts

text: the

text: position

text: and

text: the

text: width

text: of

text: the

text: eﬀ

text: tations,

text: therefore

text: the

text: eigenstates

text: can

text: be

text: written

text: as

text: resonance.

text: Therefore

text: to

text: avoid

text: numerical

text: artifacts

text: and

text: eimθ(ψ

text: (r),

text: ψ

text: (r),

text: ψ−(r)).

text: For

text: each

text: value

text: of

text: m,

text: the

text: re-

text: increase

text: the

text: numerical

text: accuracy,

text: we

text: replace

text: the

text: antitrap-

text: H

text: only

text: depends

text: on

text: the

text: radial

text: coor-

text: ping

text: potential

text: for

text: ψ−

text: by

text: m

text: dinate

text: r.

text: Since

text: the

text: value

text: of

text: the

text: functions

text: ψ

text: (r)

text: at

text: r

text: j

text: is

text: not

text: ﬁxed

text: by

text: any

text: boundary

text: condition,

text: one

text: uses

text: a

text: dis-

text: cretization

text: scheme

text: that

text: does

text: not

text: contain

text: the

text: point

text: r

text: V

text: (r)

text: ρ2

text: scatt

text: ρ2

text: namely

text: the

text: grid

text: points

text: are

text: r

text: for

text: n

text: rang-

text: n

text: ing

text: from

text: to

text: a

text: maximum

text: value

text: N.

text: ﬁxes

text: the

text: number

text: where

text: h

text: is

text: a

text: small

text: parameter,

text: typically

text: h

text: For

text: of

text: grid

text: points

text: per

text: harmonic

text: length,

text: whereas

text: N∆r

text: corre-

text: hr

text: then

text: V

text: (r)

text: V

text: (r),

text: whereas

text: for

text: hr

text: then

text: sponds

text: to

text: the

text: size

text: of

text: the

text: system

text: in

text: harmonic

text: length

text: units.

text: scatt

text: V

text: (r)

text: such

text: that

text: the

text: scattering

text: states

text: have

text: The

text: Hamiltonian

text: H

text: is

text: Hermitian

text: for

text: the

text: scalar

text: product

text: scatt

text: m

text: a

text: well

text: deﬁned

text: wavelength

text: for

text: large

text: r

text: values.

text: We

text: have

text: F

text: G

text: rdr

text: f

text: (r)g

text: (r)

text: f

text: (r)g

text: (r)

text: f−(r)g−(r)

text: numerically

text: checked

text: that

text: the

text: results

text: presented

text: here

text: are

text: where

text: F

text: (f

text: (r),

text: f

text: (r),

text: f−(r)),

text: such

text: that

text: it

text: is

text: the

text: dis-

text: insensitive

text: to

text: the

text: actual

text: h

text: value.

text: cretization

text: of

text: the

text: equation

text: rH

text: ψ

text: Erψ

text: that

text: leads

text: to

text: m

text: a

text: generalized

text: eigenvalue

text: problem

text: AX

text: EBX,

text: where

text: A

text: and

text: B

text: are

text: Hermitian

text: matrices,

text: B

text: C.

text: Results

text: being

text: positive

text: deﬁnite.

text: Neglecting

text: the

text: oﬀ-diagonal

text: coupling,

text: the

text: eﬀective

text: po-

text: tentials

text: seen

text: by

text: ψ

text: and

text: ψ−

text: components

text: being

text: and

text: The

text: properties

text: of

text: the

text: resonances

text: (position

text: in

text: energy

text: and

text: width)

text: are

text: displayed,

text: as

text: functions

text: of

text: ω

text: /ω

text: by

text: the

text: V

text: (r),

text: the

text: corresponding

text: eigenstates

text: are

text: not

text: bound

text: T

text: L

text: Fig.

text: for

text: the

text: m

text: states

text: and

text: by

text: the

text: Fig.

text: for

text: the

text: states

text: but

text: scattering

text: states,

text: whereas

text: they

text: are

text: bound

text: m

text: states.

text: In

text: both

text: cases,

text: the

text: zero

text: of

text: energy

text: cor-

text: states

text: for

text: the

text: ψ

text: component.

text: The

text: oﬀ-diagonal

text: coupling

text: responds

text: to

text: the

text: bottom

text: of

text: the

text: trapping

text: state,

text: i.e.,

text: cor-

text: allows

text: this

text: bound

text: states

text: to

text: decay

text: to

text: the

text: scattering

text: chan-

text: responding

text: to

text: a

text: global

text: shift

text: of

text: of

text: the

text: eigenergies

text: of

text: nels,

text: which,

text: from

text: a

text: mathematical

text: point

text: of

text: view,

text: becomes

text: ρ2

text: resonances,

text: i.e−.,

text: complex

text: poles

text: of

text: the

text: Green

text: function

text: Heﬀ,

text: such

text: that

text: in

text: the

text: adiabatic

text: limit

text: ρ

text: the

text: ener-

text: G(z)

text: H).

text: The

text: complex

text: rotation

text: method

text: is

text: gies

text: directly

text: correspond

text: to

text: the

text: harmonic

text: oscillator

text: levels.

text: appropriate

text: to

text: compute

text: directly

text: the

text: properties

text: of

text: these

text: This

text: is

text: clearly

text: seen≈on

text: the

text: left

text: part

text: of

text: the

text: top

text: plots.

text: On

text: resonances

text: (energy,

text: width).

text: Its

text: properties

text: rely

text: on

text: math-

text: the

text: contrary,

text: for

text: ρ

text: (right

text: part),

text: the

text: energy

text: levels

text: dif-

text: ematical

text: properties

text: of

text: the

text: analytic

text: continuation

text: of

text: the

text: fer

text: from

text: the

text: ha−rmonic

text: one,

text: in

text: particular,

text: the

text: diﬀerence

text: Green’s

text: function

text: in

text: the

text: complex

text: plane

text: A

text: review

text: in

text: energy

text: En

text: gets

text: smaller

text: with

text: higher

text: n

text: reﬂecting

text: of

text: its

text: application

text: to

text: atomic

text: physics

text: can

text: be

text: found

text: in

text: the

text: linear

text: behavior

text: of

text: the

text: trapping

text: potential

text: V

text: (r)

text: at

text: large

text: The

text: method

text: is

text: implemented

text: in

text: our

text: case

text: by

text: making

text: the

text: r.

text: The

text: behavior

text: in

text: the

text: adiabatic

text: regime

text: can

text: be

text: obtained

text: radial

text: coordinate

text: complex:

text: r

text: eiφr

text: and

text: e

text: iφ

text: from

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text: 7 [8] J. Dalibard, F. Gerbier, G. Juzeliunas, P. O¨hberg, and I. B. S[1p3i]elJm.aSnchRmeive.dMmoayde.rPahnyds.A8.3S, c1r5in23zi,(2Q0u1a1n).tum Semiclass. Opt. 8, 693 (1996). [9] N. Goldman, G. Juzeliunas, and P. O¨hberg, Rep. Prog. Phys[.174]7,E1.2B64a0ls1le(v20a1n4d).J. M. Combes, Commun. Math. Phys. 22, 280 (1971). [10] A. Buchleitner, B. Gr´emaud, and D. Delande, J. Phys. B: At[.15M] oYl..OKp.tH. Po,hPysh.y2s.7R, 2e6p6.399(1,9194(1).983). [11] J. Ruseckas, G. Juzeliunas, P. O¨hberg, and M. Fleischauer, P[1h6y]s.AR.eGv.. LMeatrtt.i9n5, ,K0.1H04e0lm4e(r2s0o0n5,)V. . S. Bagnato, G. P. Lafyatis and D. E. Pritchard, Phys. Rev. Lett. 61, 2431 (1988). [12] Y. -J. Lin, K. Jim´enez-Garc´ıa, and I. B. Spielman, Nature (London), 471, 83 (2011).

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text: Statis-

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text: Bednorz,

text: W.

text: Belzig

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text: A.

text: Nitzan,

text: New

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text: Phys.

text: tics,

text: and

text: All

text: that

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text: New

text: York,

text: N.

text: Bogoliubov,

text: A.A.

text: Logunov,

text: A.I.

text: Oksak,

text: and

text: I.T.

text: A.

text: Bednorz,

text: K.

text: Franke,

text: and

text: W.

text: Belzig,

text: New

text: J.

text: Phys.

text: Todorov,

text: General

text: Principles

text: of

text: Quantum

text: Field

text: Theory,

text: (Kluwer

text: Academic

text: Publishers,

text: Dordrecht,

text: L.H.

text: E.P.

text: Wigner,

text: Phys.

text: Rev.

text: M.

text: Hillery

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text: al.,

text: Phys.

text: Rep.

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text: Weinberg,

text: The

text: Quantum

text: A.

text: Bednorz,

text: New

text: J.

text: Phys.

text: Theory

text: of

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text: Cam-

text: A.A.

text: Clerk,

text: M.H.

text: Devoret,

text: S.M.

text: Girvin,

text: F.

text: Marquardt,

text: bridge,

text: M.

text: Peskin,

text: D.

text: Schroeder,

text: An

text: Introduc-

text: and

text: R.J.

text: Schoelkopf,

text: Rev.

text: Mod.

text: Phys.

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text: to

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text: (Perseus

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text: C.

text: Bruder,

text: B.

text: Reulet

text: and

text: W.

text: Belzig,

text: Phys.

text: J.D.

text: Bjorken

text: and

text: S.D.

text: Drell,

text: Relativistic

text: Quantum

text: Rev.

text: Lett.

text: Mechanics

text: (McGraw-Hill,

text: New

text: York,

text: R.

text: Zwanzig,

text: Physica

text: A.

text: Bednorz,

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text: Phys.

text: J.

text: C

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text: Schwinger,

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text: Phys.

text: L.P.

text: Kadanoﬀ

text: H.M.

text: Wiseman

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text: G.J.

text: Milburn,

text: Quantum

text: Measure-

text: and

text: G.

text: Baym,

text: Quantum

text: Statistical

text: Mechanics,

text: (W.A.

text: ment

text: and

text: Control

text: (Cambridge

text: University

text: Press,

text: Cam-

text: Benjamin,

text: New

text: York,

text: L.

text: Keldysh,

text: Sov.

text: Phys.

text: bridge,

text: England,

text: JETP

text: K.

text: Kraus,

text: States,

text: Eﬀects

text: and

text: Operations

text: (Springer,

text: K.

text: Chou,

text: Z.

text: Su,

text: B.

text: Hao

text: and

text: L.

text: Yu,

text: Phys.

text: Rep.

text: Berlin,

text: N.P.

text: Landsman

text: and

text: C.G.

text: van

text: Weert,

text: Phys.

text: Rep.

text: G.C.

text: Ghirardi,

text: A.

text: Rimini

text: and

text: T.

text: Weber,

text: Phys.

text: Rev.

text: D

text: A.

text: Bednorz,

text: W.

text: Bednorz,

text: and

text: W.

text: Belzig,

text: Phys.

text: Rev.

text: A

text: P.

text: Pearle,

text: Phys.

text: Rev.

text: A

text: G.C.

text: Ghirardi,

text: P.

text: Pearle,

text: and

text: A.

text: Rimini,

text: Phys.

text: Rev.

text: A

text: H.B.

text: Callen

text: and

text: T.A.

text: Welton,

text: Phys.

text: Rev.

text: G.C.

text: Ghirardi,

text: R.

text: Grassi,

text: and

text: P.

text: Pearle,

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text: Phys.

text: R.

text: Karplus

text: and

text: M.

text: Neuman,

text: Phys.

text: Rev.

text: A.

text: Kostelecky

text: and

text: N.

text: Russell,

text: Rev.

text: Mod.

text: Phys.

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text: Aharonov,

text: D.Z.

text: Albert

text: and

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text: Vaidman,

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text: Rev.

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text: H.

text: Wiseman,

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text: 3 Im t work is to check if such a construction is possible to in- Re t clude free choice. →Namely, all readouts will be choice iβ conditioned, e.g. o o , o . This means that readouts (a) (b) a ab for all choices, also those not just realized, are measur- able. One can extend this idea naturally to continuous ﬁelds and choices, and then α[x, a] is a function of x and FIG. 2: Schwinger-Keldysh time contour (a) in general with functional of a. We assume that N is an independent the left part unspeciﬁed and (b) for a thermal state with β = choice and state of the system. Otherwise we would have 1/k T. The shape of the left line is arbitrary. The time B additional choice-controlled dynamics. In that case we window for observations is bounded by the horizontal part will rather incorporate all such dependence in the quan- tum description alone. This is a reasonable minimalist approach, where quantum mechanics essentially captures Fig.2(b). It is important to discriminate between for- − all the dynamics. ward, +i(cid:15)0 , and backward, i(cid:15), times x0±, respectively, → The quasiprobability statistics can be conveniently with (cid:15) 0+ (±the sp→atial position is unaﬀected) . In such written in the form for correlations [17, 21], description Oˇ (x) O(x±) and time order is dropped (except the fact that fermion ﬁelds are anticommuting (cid:90) (cid:104) · · · (cid:105) Grassmann numbers). Free ﬁeld a(x±) = a(x) is the o1(x1) on(xn) Q = same for forward and backward time. (cid:48) T (cid:104) − (cid:48) (cid:48) · · · − (cid:48) (cid:48) (cid:105) To proceed with the problem of relativistic invariant dnx Oˇnxn xn(xn) Oˇ1x1 x1(x1) . (5) realism, we have to recall the relativistic framework. We shall use standard relativistic quantum ﬁeld notation T (cid:48) with four-vectors Aµ (e.g., ﬁeld); xµ (position in space- where denotes time ordering, with respect to x 0, (cid:80) − − − time); a ﬂat metric gµν = g = diag(1, 1, 1, 1); and µν · summation convention and index shifting X Y = XµY = XµY = Xµg Y ν = X gµνY , Xµ = − (cid:48) (cid:48) − (cid:48) (cid:48) − (cid:48) (cid:48) µ µ µ µν µ ν Oˇx x (x ) = δ(x x )Oˇc(x ) + f(x x )Oˇq(x )/2 . (6) gµνXν, Xµ = gµνXν, with derivatives ∂µ = ∂/∂xµ. Along the Schwinger-Keldysh contour we parametrize (cid:82) x0(s) by real s with dx0 = (dx0/ds)ds and The superoperators Oˇc/q [22] act on any operator Xˆ ∂ = (dx0/ds)−1∂ . We shall often switch to momen- { } 0 s · as an anticommutator/commutator: OˇcXˆ = Oˆ, Xˆ /2 tum or Fourier space with X(p) = d4xeip xX(x), which − − and iOˇqXˆ = [Oˆ, Xˆ]. Alternatively 2Oˇc = Oˇ+ + Oˇ and needs us to specify x along either the + or part. Then iOˇq = Oˇ+−Oˇ− with Oˇ+Xˆ = OˆXˆ and Oˇ−Xˆ = XˆOˆ. The the equilibrium f gives Fourier transform f(p) = i sgnp0 [21]. However, if we want relativistic invariance, the function f is in principle arbitrary but it turns out that · proper choice is f(p) = i sgnp0θ(p p) [19]. In any reason- only two choices are reasonable, in particular f = 0 (no · memory) [15, 17] or f(x) = δ3((cid:126)x)/πx0 (no correlations in able choice we have f(p→) = 0 for spacelike p, p p < 0. It zero temperature equilibrium) [21]. The operators Oˆ are essentially means Oˇ(p) →(O+(p) + O−(p))/2 for space- like p or f = 0 and Oˇ(p) O±(p) for the other f and given in the Heisenberg picture including the free part · ± timelike p, p p > 0, with p0 > 0. The invariance of Q governed by the ﬁeld a. In principle in (5) one could de- (cid:48) follows then from the Wightman axiom (3), so it remains ﬁne correlations for diﬀerent a and a (or more) but they to check if N also can be invariant. are not directly measurable. For our goal it is suﬃcient IV. AN ATTEMPT OF INVARIANT REALISM to consider a single a. In quantum ﬁeld theory, the above can be written in terms of path integ(cid:90)rals, name(cid:90)ly A simple convolution with positive N makes it impos- sible to construct relativistic invariant realism even with- out free choice, because of zero-temperature counterex- (cid:90) (cid:90) (cid:104) (cid:105) L X Dφ exp id4x (φ, ∂φ) amples [19]. However we can avoid the zero-temperature problems by simply subtracting zero-temperature statis- L tics. It can be achieved in the following way. The con- = Dφ exp id4x (φ, ∂φ)X (7) volution (4) is equivalent to a simple sum of generating functions, namely, L with Lagrangian density and integration over x0 (cid:90) (cid:90) along the Schwinger-Keldysh-Kadanoﬀ-Baym contour S [χ] = S [χ] + S [χ], [23, 24], shown in Fig. 2(a), where the state is described P N Q L by properly deﬁning and the path of x0 before the eSX[χ] = DoX[o] exp id4xo(x)χ(x). (8) earliest x0 with an active choice or observation. For instance a thermal state of temperature T means sim- ply extending x0 to complex values with a jump of iβ Generating functions can be used as a formal series ∞ → (β = 1/k T, becomes i at T 0 ) as shown in with cumulant expansion in χ, e.g., B + ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Aµ

text: (e.g.,

text: ﬁeld);

text: xµ

text: (position

text: in

text: space-

text: where

text: denotes

text: time

text: ordering,

text: with

text: respect

text: to

text: x

text: time);

text: a

text: ﬂat

text: metric

text: gµν

text: g

text: and

text: µν

text: summation

text: convention

text: and

text: index

text: shifting

text: X

text: Y

text: XµY

text: XµY

text: Xµg

text: Y

text: ν

text: X

text: gµνY

text: Xµ

text: µ

text: µ

text: µ

text: µν

text: µ

text: ν

text: Oˇx

text: x

text: (x

text: δ(x

text: x

text: )Oˇc(x

text: f(x

text: x

text: )Oˇq(x

text: gµνXν,

text: Xµ

text: gµνXν,

text: with

text: derivatives

text: ∂µ

text: ∂/∂xµ.

text: Along

text: the

text: Schwinger-Keldysh

text: contour

text: we

text: parametrize

text: by

text: real

text: s

text: with

text: and

text: The

text: superoperators

text: Oˇc/q

text: act

text: on

text: any

text: operator

text: Xˆ

text: We

text: shall

text: often

text: switch

text: to

text: momen-

text: s

text: as

text: an

text: anticommutator/commutator:

text: OˇcXˆ

text: Oˆ,

text: Xˆ

text: tum

text: or

text: Fourier

text: space

text: with

text: X(p)

text: xX(x),

text: which

text: and

text: iOˇqXˆ

text: [Oˆ,

text: Xˆ].

text: Alternatively

text: Oˇ+

text: Oˇ

text: and

text: needs

text: us

text: to

text: specify

text: x

text: along

text: either

text: the

text: or

text: part.

text: Then

text: iOˇq

text: Oˇ+−Oˇ−

text: with

text: Oˇ+Xˆ

text: OˆXˆ

text: and

text: Oˇ−Xˆ

text: XˆOˆ.

text: The

text: the

text: equilibrium

text: f

text: gives

text: Fourier

text: transform

text: f(p)

text: i

text: However,

text: if

text: we

text: want

text: relativistic

text: invariance,

text: the

text: function

text: f

text: is

text: in

text: principle

text: arbitrary

text: but

text: it

text: turns

text: out

text: that

text: proper

text: choice

text: is

text: f(p)

text: i

text: p)

text: In

text: any

text: reason-

text: only

text: two

text: choices

text: are

text: reasonable,

text: in

text: particular

text: f

text: (no

text: memory)

text: or

text: f(x)

text: (no

text: correlations

text: in

text: able

text: choice

text: we

text: have

text: f(p→)

text: for

text: spacelike

text: p,

text: p

text: p

text: It

text: zero

text: temperature

text: equilibrium)

text: The

text: operators

text: Oˆ

text: are

text: essentially

text: means

text: Oˇ(p)

text: →(O+(p)

text: for

text: space-

text: like

text: p

text: or

text: f

text: and

text: Oˇ(p)

text: O±(p)

text: for

text: the

text: other

text: f

text: and

text: given

text: in

text: the

text: Heisenberg

text: picture

text: including

text: the

text: free

text: part

text: timelike

text: p,

text: p

text: p

text: with

text: The

text: invariance

text: of

text: Q

text: governed

text: by

text: the

text: ﬁeld

text: a.

text: In

text: principle

text: in

text: one

text: could

text: de-

text: follows

text: then

text: from

text: the

text: Wightman

text: axiom

text: so

text: it

text: remains

text: ﬁne

text: correlations

text: for

text: diﬀerent

text: a

text: and

text: a

text: (or

text: more)

text: but

text: they

text: to

text: check

text: if

text: N

text: also

text: can

text: be

text: invariant.

text: are

text: not

text: directly

text: measurable.

text: For

text: our

text: goal

text: it

text: is

text: suﬃcient

text: IV.

text: AN

text: ATTEMPT

text: OF

text: INVARIANT

text: REALISM

text: to

text: consider

text: a

text: single

text: a.

text: In

text: quantum

text: ﬁeld

text: theory,

text: the

text: above

text: can

text: be

text: written

text: in

text: terms

text: of

text: path

text: A

text: simple

text: convolution

text: with

text: positive

text: N

text: makes

text: it

text: impos-

text: sible

text: to

text: construct

text: relativistic

text: invariant

text: realism

text: even

text: with-

text: out

text: free

text: choice,

text: because

text: of

text: zero-temperature

text: counterex-

text: L

text: X

text: Dφ

text: exp

text: (φ,

text: ∂φ)

text: amples

text: However

text: we

text: can

text: avoid

text: the

text: zero-temperature

text: problems

text: by

text: simply

text: subtracting

text: zero-temperature

text: statis-

text: L

text: tics.

text: It

text: can

text: be

text: achieved

text: in

text: the

text: following

text: way.

text: The

text: con-

text: Dφ

text: exp

text: (φ,

text: ∂φ)X

text: volution

text: is

text: equivalent

text: to

text: a

text: simple

text: sum

text: of

text: generating

text: functions,

text: namely,

text: L

text: with

text: Lagrangian

text: density

text: and

text: integration

text: over

text: along

text: the

text: Schwinger-Keldysh-Kadanoﬀ-Baym

text: contour

text: S

text: [χ]

text: S

text: [χ]

text: S

text: [χ],

text: shown

text: in

text: Fig.

text: where

text: the

text: state

text: is

text: described

text: P

text: N

text: Q

text: L

text: by

text: properly

text: deﬁning

text: and

text: the

text: path

text: of

text: before

text: the

text: eSX[χ]

text: DoX[o]

text: exp

text: earliest

text: with

text: an

text: active

text: choice

text: or

text: observation.

text: For

text: instance

text: a

text: thermal

text: state

text: of

text: temperature

text: T

text: means

text: sim-

text: ply

text: extending

text: to

text: complex

text: values

text: with

text: a

text: jump

text: of

text: iβ

text: Generating

text: functions

text: can

text: be

text: used

text: as

text: a

text: formal

text: series

text: (β

text: T,

text: becomes

text: i

text: at

text: T

text: as

text: shown

text: in

text: with

text: cumulant

text: expansion

text: in

text: χ,

text: e.g.,

text: B

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 dδ cγ sumption of every Bell test [1–3] and therefore it is tested there simultaneously. Although in recent experiments [4– 6] the signature of superluminal signaling based on the reported data seems to be yet insigniﬁcant, in all of them both random choices and readouts are machine made so aα bβ fair time tagging and choice is a matter of trust in elec- tronics, not humans – the choice is not free in the human sense [29]. Further and improved experiments should be FIG. 5: One of three (or six when directions are counted) continued to resolve the question of possible superluminal diagrams T contributing to G. Gamma matrices are inserted signaling. in vertices and propagators in lines VII. CONCLUSIONS The presented direct conﬂicts of freedom of choice · in quantum realism with relativity demonstrates incom- where the minus is due to anticommuting, →p/ = γ p, pleteness of the present quantum framework without us- the factor 2 due to opposite directions and (cid:15) 0 due + ing the assumption of the Bell theorem. The easiest way to the limits of x0. It is important that G (not T) be seems to abandon relativistic invariance. This can be invariant with respect to the permutation of pairs (a, α), tested experimentally, especially by no-signaling in the (b, β), (c, γ), (d, δ); relativistically invariant; and gauge test of local realism, which is diﬀerent from the direct invariant, namely a Gαβγδ(abcd) = 0 (and analogously α search for violations of relativistic invariance [28]. The- for other pairs). It is easily proved by the identity oretical and experimental development of such tests is critical for ﬁnding a way to reconcile quantum realism − − − − − with free choices. Finally, the freedom of choice remains (p/ m + i(cid:15)) 1 (p/ + a/ m + i(cid:15)) 1 = − − − − a matter of trust in electronics, with human choice yet (p/ m + i(cid:15)) 1a/(p/ + a/ m + i(cid:15)) 1 (32) to be considered [29]. ACKNOWLEDGMENTS used at every occurrence of a/, and by telescoping the cancellation of the left-hand sides from all parts of the W. Belzig, R. Demkowicz-Dobrzan´ski, and P. integral, with a shift of p when appropriate. Chankowski are acknowledged for motivation, discussion, From relativistic invariance G must consist of three and suggestions. types of terms, kαqβrγsδ (heads), gαβrγsδ (and permu- APPENDIX FOUR-POINT GREEN FUNCTION tations), and gαβgγδ (and two other permutations) mul- tiplied by scalar functions of abcd. Here kqrs are equal to We shall recall the calculation of the four-point elec- some of abcd but from the condition a + b + c + d = 0 we tron Green function [27] deﬁned as follows: can exclude a from k, b from q, c from r, and d from s (by − − − substituting a = b c d, etc.). From gauge invariance (cid:104) (cid:105) Gαβγδ(x x x x ) = jα(x )jβ (x )jγ (x )jδ (x ) . (29) the heads determine all other terms, because the terms a b c d + a + b + c + d with g cannot exist without heads. It is clear when e.g. we combine gαβrγsδ with a , which gives aβrγsδ. With- (cid:82) α We will rather refe(cid:104)r to the Fourier tr(cid:105)ansform of out heads this term can be canceled(cid:54) only by gαγqβsδ or G(a, b, c, d) equal to j+(a)j+(b)j+(c)j+·(d) with the gαδqβr↔γ. This is impossible if r, s = a. By interchang- Fourier transform j(a) = d4xj(xa)eixa a. Thanks to ing α β we ﬁnd that rs must correspond to ab or translational invariance G = δ(a + b + c + d)G˜. The cal- ba. However, then gαγqβsδ implies q = c and gαδqβrγ culation of G˜ by standard methods (Wick decomposition implies q = d, which again makes the cancellation im- possible. Terms gαβgγδ left alone have nothing to cancel into propagators – two-point Green functions) reduces to with. Heads can be classiﬁed into six types: three integrals (diﬀering by permutation), corresponding to the box Feynman diagram depicted in Fig. 5: 1 : bαaβdγcδ, 2 : dαaβbγcδ, 3 : bαaβaγaδ, (33) G˜αβγδ(abcd) = Tαβγδ(abcd) + Tβαγδ(bacd)Tδβγα(dbca) 4 : bαaβbγaδ, 5 : bαcβaγaδ, 6 : bαaβdγaδ, (30) with (cid:90) depicted in Fig. 6. Gauge invariance should allow us to write − − − − × H = A B C D Gαβγδ in terms of ﬁelds A = ∂ A Tαβγδ(abcd) = 2 d4pTr(p/ m + i(cid:15)) 1γα ∂ A oαr iβn tγhe δFourier representation i(a µAν − µa Aν ) ν µ ν µ µ ν − − − − × (similarly for BCD). In each combination we get some (p/ + a/ m + i(cid:15)) 1γβ(p/ + a/ + /b m + i(cid:15)) 1 heads There are plenty of possible types and combina- − − γγ(p/ d/ m + i(cid:15))γδ, (31) tions of ﬁelds that may contribute to H, but we shall ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dδ

text: cγ

text: sumption

text: of

text: every

text: Bell

text: test

text: and

text: therefore

text: it

text: is

text: tested

text: there

text: simultaneously.

text: Although

text: in

text: recent

text: experiments

text: the

text: signature

text: of

text: superluminal

text: signaling

text: based

text: on

text: the

text: reported

text: data

text: seems

text: to

text: be

text: yet

text: insigniﬁcant,

text: in

text: all

text: of

text: them

text: both

text: random

text: choices

text: and

text: readouts

text: are

text: machine

text: made

text: so

text: aα

text: bβ

text: fair

text: time

text: tagging

text: and

text: choice

text: is

text: a

text: matter

text: of

text: trust

text: in

text: elec-

text: tronics,

text: not

text: humans

text: the

text: choice

text: is

text: not

text: free

text: in

text: the

text: human

text: sense

text: Further

text: and

text: improved

text: experiments

text: should

text: be

text: FIG.

text: One

text: of

text: three

text: (or

text: six

text: when

text: directions

text: are

text: counted)

text: continued

text: to

text: resolve

text: the

text: question

text: of

text: possible

text: superluminal

text: diagrams

text: T

text: contributing

text: to

text: G.

text: Gamma

text: matrices

text: are

text: inserted

text: signaling.

text: in

text: vertices

text: and

text: propagators

text: in

text: lines

text: VII.

text: CONCLUSIONS

text: The

text: presented

text: direct

text: conﬂicts

text: of

text: freedom

text: of

text: choice

text: in

text: quantum

text: realism

text: with

text: relativity

text: demonstrates

text: incom-

text: where

text: the

text: minus

text: is

text: due

text: to

text: anticommuting,

text: →p/

text: γ

text: p,

text: pleteness

text: of

text: the

text: present

text: quantum

text: framework

text: without

text: us-

text: the

text: factor

text: due

text: to

text: opposite

text: directions

text: and

text: due

text: ing

text: the

text: assumption

text: of

text: the

text: Bell

text: theorem.

text: The

text: easiest

text: way

text: to

text: the

text: limits

text: of

text: It

text: is

text: important

text: that

text: G

text: (not

text: T)

text: be

text: seems

text: to

text: abandon

text: relativistic

text: invariance.

text: This

text: can

text: be

text: invariant

text: with

text: respect

text: to

text: the

text: permutation

text: of

text: pairs

text: (a,

text: α),

text: tested

text: experimentally,

text: especially

text: by

text: no-signaling

text: in

text: the

text: (b,

text: β),

text: (c,

text: γ),

text: (d,

text: δ);

text: relativistically

text: invariant;

text: and

text: gauge

text: test

text: of

text: local

text: realism,

text: which

text: is

text: diﬀerent

text: from

text: the

text: direct

text: invariant,

text: namely

text: a

text: Gαβγδ(abcd)

text: (and

text: analogously

text: α

text: search

text: for

text: violations

text: of

text: relativistic

text: invariance

text: The-

text: for

text: other

text: pairs).

text: It

text: is

text: easily

text: proved

text: by

text: the

text: identity

text: oretical

text: and

text: experimental

text: development

text: of

text: such

text: tests

text: is

text: critical

text: for

text: ﬁnding

text: a

text: way

text: to

text: reconcile

text: quantum

text: realism

text: with

text: free

text: choices.

text: Finally,

text: the

text: freedom

text: of

text: choice

text: remains

text: (p/

text: m

text: (p/

text: a/

text: m

text: a

text: matter

text: of

text: trust

text: in

text: electronics,

text: with

text: human

text: choice

text: yet

text: (p/

text: m

text: a/

text: m

text: to

text: be

text: considered

text: ACKNOWLEDGMENTS

text: used

text: at

text: every

text: occurrence

text: of

text: a/,

text: and

text: by

text: telescoping

text: the

text: cancellation

text: of

text: the

text: left-hand

text: sides

text: from

text: all

text: parts

text: of

text: the

text: W.

text: Belzig,

text: R.

text: Demkowicz-Dobrzan´ski,

text: and

text: P.

text: integral,

text: with

text: a

text: shift

text: of

text: p

text: when

text: appropriate.

text: Chankowski

text: are

text: acknowledged

text: for

text: motivation,

text: discussion,

text: From

text: relativistic

text: invariance

text: G

text: must

text: consist

text: of

text: three

text: and

text: suggestions.

text: types

text: of

text: terms,

text: kαqβrγsδ

text: (heads),

text: gαβrγsδ

text: (and

text: permu-

text: APPENDIX

text: FOUR-POINT

text: GREEN

text: FUNCTION

text: tations),

text: and

text: gαβgγδ

text: (and

text: two

text: other

text: permutations)

text: mul-

text: tiplied

text: by

text: scalar

text: functions

text: of

text: abcd.

text: Here

text: kqrs

text: are

text: equal

text: to

text: We

text: shall

text: recall

text: the

text: calculation

text: of

text: the

text: four-point

text: elec-

text: some

text: of

text: abcd

text: but

text: from

text: the

text: condition

text: a

text: b

text: c

text: d

text: we

text: tron

text: Green

text: function

text: deﬁned

text: as

text: follows:

text: can

text: exclude

text: a

text: from

text: k,

text: b

text: from

text: q,

text: c

text: from

text: r,

text: and

text: d

text: from

text: s

text: (by

text: substituting

text: a

text: b

text: c

text: d,

text: etc.).

text: From

text: gauge

text: invariance

text: Gαβγδ(x

text: x

text: x

text: x

text: jα(x

text: )jβ

text: (x

text: )jγ

text: (x

text: )jδ

text: (x

text: the

text: heads

text: determine

text: all

text: other

text: terms,

text: because

text: the

text: terms

text: a

text: b

text: c

text: d

text: a

text: b

text: c

text: d

text: with

text: g

text: cannot

text: exist

text: without

text: heads.

text: It

text: is

text: clear

text: when

text: e.g.

text: we

text: combine

text: gαβrγsδ

text: with

text: a

text: which

text: gives

text: aβrγsδ.

text: With-

text: α

text: We

text: will

text: rather

text: to

text: the

text: Fourier

text: of

text: out

text: heads

text: this

text: term

text: can

text: be

text: only

text: by

text: gαγqβsδ

text: or

text: G(a,

text: b,

text: c,

text: d)

text: equal

text: to

text: j+(a)j+(b)j+(c)j+·(d)

text: with

text: the

text: gαδqβr↔γ.

text: This

text: is

text: impossible

text: if

text: r,

text: s

text: a.

text: By

text: interchang-

text: Fourier

text: transform

text: j(a)

text: a.

text: Thanks

text: to

text: ing

text: α

text: β

text: we

text: ﬁnd

text: that

text: rs

text: must

text: correspond

text: to

text: ab

text: or

text: translational

text: invariance

text: G

text: δ(a

text: b

text: c

text: d)G˜.

text: The

text: cal-

text: ba.

text: However,

text: then

text: gαγqβsδ

text: implies

text: q

text: c

text: and

text: gαδqβrγ

text: culation

text: of

text: G˜

text: by

text: standard

text: methods

text: (Wick

text: decomposition

text: implies

text: q

text: d,

text: which

text: again

text: makes

text: the

text: cancellation

text: im-

text: possible.

text: Terms

text: gαβgγδ

text: left

text: alone

text: have

text: nothing

text: to

text: cancel

text: into

text: propagators

text: two-point

text: Green

text: functions)

text: reduces

text: to

text: with.

text: Heads

text: can

text: be

text: classiﬁed

text: into

text: six

text: types:

text: three

text: integrals

text: (diﬀering

text: by

text: permutation),

text: corresponding

text: to

text: the

text: box

text: Feynman

text: diagram

text: depicted

text: in

text: Fig.

text: bαaβdγcδ,

text: dαaβbγcδ,

text: bαaβaγaδ,

text: G˜αβγδ(abcd)

text: Tαβγδ(abcd)

text: Tβαγδ(bacd)Tδβγα(dbca)

text: bαaβbγaδ,

text: bαcβaγaδ,

text: bαaβdγaδ,

text: with

text: depicted

text: in

text: Fig.

text: Gauge

text: invariance

text: should

text: allow

text: us

text: to

text: write

text: H

text: A

text: B

text: C

text: D

text: Gαβγδ

text: in

text: terms

text: of

text: ﬁelds

text: A

text: A

text: Tαβγδ(abcd)

text: m

text: 1γα

text: A

text: oαr

text: iβn

text: tγhe

text: δFourier

text: representation

text: i(a

text: µAν

text: µa

text: Aν

text: ν

text: µ

text: ν

text: µ

text: µ

text: ν

text: (similarly

text: for

text: BCD).

text: In

text: each

text: combination

text: we

text: get

text: some

text: (p/

text: a/

text: m

text: a/

text: /b

text: m

text: heads

text: There

text: are

text: plenty

text: of

text: possible

text: types

text: and

text: combina-

text: γγ(p/

text: d/

text: m

text: tions

text: of

text: ﬁelds

text: that

text: may

text: contribute

text: to

text: H,

text: but

text: we

text: shall

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 − − − − 32bαaβdγcδλ λ (1 λ )(1 λ ), 32dαcβbγaδλ λ (1 λ )(1 λ ), ab cd ab cd bc da bc da 32cαdβaγbδ(λ + λ )(λ + λ )(λ + λ )(λ + λ ), (41) ab bc bc cd cd da da ab and type 2 is − − − − 4(bαcβdγaδ + dαaβbγcδ)(((1 λ )(1 λ ) + λ λ )((1 λ )(1 λ ) + λ λ ) da cd cd da ab bc ab bc − − − − − +((1 λ )λ + (1 λ )λ )((1 λ )λ + (1 λ )λ )) 4(cαaβdγbδ + bαdβaγcδ)(((λ + λ )(λ + λ ) da cd cd da bc ab ab bc bc cd cd da − − +(λ + λ )(λ + λ ))(λ λ + (1 λ )(1 λ )) + ((λ + λ )(λ + λ ) + (λ + λ )(λ + λ )) da ab ab bc ab cd ab cd cd da da ab ab bc bc cd × − − (λ (1 λ ) + λ (1 λ ))), (42) ab cd cd ab − − − 4(dαcβaγbδ + cαdβbγaδ)(((λ + λ )(λ + λ ) + (λ + λ )(λ + λ ))(λ λ + (1 λ )(1 λ )) cd da da ab ab bc bc cd bc da bc da − − +((λ + λ )(λ + λ ) + (λ + λ )(λ + λ ))(λ (1 λ ) + λ (1 λ ))), bc cd cd da da ab ab bc bc da da bc appear) or other groups (we cannot get the same cycles). 5 · · · 1 1 1 Hence R(a d)+S(b d)+T(c d) = 0 and we can rewrite · · − the group multiplied by (c d) in the form (R((c d)aδ · · − · − (a d)cδ)+S((c d)bδ (b d)cδ))(bαcβaγ cαaβbγ). which corresponds to a combination of tensors 5. → Making the Wick rotation p0 ip0, we can also inte- 2 2 2 5 gra(cid:90)te (37) over p, (cid:90) ∞ (cid:90) d4p 2π2iP3dP · − = − − FIG. 7: Three contributions to the head types 1 and 2 and (p p m2 + Q + i(cid:15))4 (P2 + m2 Q i(cid:15))4 two to type 5, as in Fig. 6, inscribed in the box diagram, Fig. ∞ 0 π2iudu iπ2 5. = − − = − − . (43) (u + m2 Q i(cid:15))4 6(m2 Q i(cid:15))2 0 depicted in Fig. 7. We also ﬁnd that heads 5 appear − At small values of abcd we can neglect Q in the denom- in antisymmetric pairs, e.g., (bαcβaγ cαaβbγ)aδ and − inator of (43). It remains to integrate (41) and (42) (the (bαdβaδ dαaβbδ)aγ, depicted in Fig. 7. other tensors appear only at large values as they must To show that only tensors in (34) appear in H, note contain additional Fourier variables abcd from Q) over λ, that (a) we can take away heads 1 and 2 by tensors 1 and the ﬁnal result is and 2 leaving the rest invariant, (b) heads 3 and 4 can be taken away by tensors 3 and 4 leaving only heads 5 and 6, (c) heads 6, e.g., bαaβbγcδ, must get canceled −8iπ2 · · · · · · when multiplied by d , which cannot be achieved either H = (( A B )( C D ) + δ 9m4 by terms with g (they will contain d) or by other heads · · · · · · · · · · · · ( A C )( B D ) + ( A D )( C B )) 6 or 5 (they give diﬀerent terms). Therefore we only have to show that heads 5 alone must combine to the 28iπ2 · · · · · + (( A B C D ) + tensor 5. Let us focus ﬁrst focus on a subgroup of these 45m4 − · · · · · · · · · · heads (Raδ +Sbδ +Tcδ)(bαcβaγ cαaβbγ). Multiplied by ( A C B D ) + ( A B D C )). (44) − d cycles bαcβaγ cαaβbγ must get canceled from gauge invariance, which cannot be done by terms with g (d will [1] J.S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964); [3] P.H. Eberhard, Phys. Rev. A 47, R747 (1993). A. Shimony, plato.stanford.edu/entries/bell-theorem/. [4] B. Hensen et al., Nature (London)526, 682 (2015) [2] J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt, [5] M. Giustina et al., Phys. Rev. Lett. 115, 250401 (2015). Phys. Rev. Lett. 23, 880(1969) . [6] L.K. Shalm et al., Phys. Rev. Lett. 115, 250402 (2015). ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: ab

text: cd

text: ab

text: cd

text: bc

text: da

text: bc

text: da

text: λ

text: )(λ

text: λ

text: )(λ

text: λ

text: )(λ

text: λ

text: ab

text: bc

text: bc

text: cd

text: cd

text: da

text: da

text: ab

text: and

text: type

text: is

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: da

text: cd

text: cd

text: da

text: ab

text: bc

text: ab

text: bc

text: λ

text: )λ

text: λ

text: )λ

text: λ

text: )λ

text: λ

text: )λ

text: bαdβaγcδ)(((λ

text: λ

text: )(λ

text: λ

text: da

text: cd

text: cd

text: da

text: bc

text: ab

text: ab

text: bc

text: bc

text: cd

text: cd

text: da

text: +(λ

text: λ

text: )(λ

text: λ

text: ))(λ

text: λ

text: λ

text: λ

text: ((λ

text: λ

text: )(λ

text: λ

text: (λ

text: λ

text: )(λ

text: λ

text: da

text: ab

text: ab

text: bc

text: ab

text: cd

text: ab

text: cd

text: cd

text: da

text: da

text: ab

text: ab

text: bc

text: bc

text: cd

text: (λ

text: λ

text: λ

text: λ

text: ab

text: cd

text: cd

text: ab

text: cαdβbγaδ)(((λ

text: λ

text: )(λ

text: λ

text: (λ

text: λ

text: )(λ

text: λ

text: ))(λ

text: λ

text: λ

text: λ

text: cd

text: da

text: da

text: ab

text: ab

text: bc

text: bc

text: cd

text: bc

text: da

text: bc

text: da

text: +((λ

text: λ

text: )(λ

text: λ

text: (λ

text: λ

text: )(λ

text: λ

text: ))(λ

text: λ

text: λ

text: λ

text: bc

text: cd

text: cd

text: da

text: da

text: ab

text: ab

text: bc

text: bc

text: da

text: da

text: bc

text: appear)

text: or

text: other

text: groups

text: (we

text: cannot

text: get

text: the

text: same

text: cycles).

text: Hence

text: R(a

text: d)+S(b

text: d)+T(c

text: d)

text: and

text: we

text: can

text: rewrite

text: the

text: group

text: multiplied

text: by

text: (c

text: d)

text: in

text: the

text: form

text: (R((c

text: d)aδ

text: (a

text: d)cδ)+S((c

text: d)bδ

text: (b

text: d)cδ))(bαcβaγ

text: cαaβbγ).

text: which

text: corresponds

text: to

text: a

text: combination

text: of

text: tensors

text: Making

text: the

text: Wick

text: rotation

text: we

text: can

text: also

text: inte-

text: over

text: p,

text: FIG.

text: Three

text: contributions

text: to

text: the

text: head

text: types

text: and

text: and

text: (p

text: p

text: Q

text: Q

text: two

text: to

text: type

text: as

text: in

text: Fig.

text: inscribed

text: in

text: the

text: box

text: diagram,

text: Fig.

text: (u

text: Q

text: Q

text: depicted

text: in

text: Fig.

text: We

text: also

text: ﬁnd

text: that

text: heads

text: appear

text: At

text: small

text: values

text: of

text: abcd

text: we

text: can

text: neglect

text: Q

text: in

text: the

text: denom-

text: in

text: antisymmetric

text: pairs,

text: e.g.,

text: (bαcβaγ

text: cαaβbγ)aδ

text: and

text: inator

text: of

text: It

text: remains

text: to

text: integrate

text: and

text: (the

text: (bαdβaδ

text: dαaβbδ)aγ,

text: depicted

text: in

text: Fig.

text: other

text: tensors

text: appear

text: only

text: at

text: large

text: values

text: as

text: they

text: must

text: To

text: show

text: that

text: only

text: tensors

text: in

text: appear

text: in

text: H,

text: note

text: contain

text: additional

text: Fourier

text: variables

text: abcd

text: from

text: Q)

text: over

text: λ,

text: that

text: (a)

text: we

text: can

text: take

text: away

text: heads

text: and

text: by

text: tensors

text: and

text: the

text: ﬁnal

text: result

text: is

text: and

text: leaving

text: the

text: rest

text: invariant,

text: (b)

text: heads

text: and

text: can

text: be

text: taken

text: away

text: by

text: tensors

text: and

text: leaving

text: only

text: heads

text: and

text: (c)

text: heads

text: e.g.,

text: bαaβbγcδ,

text: must

text: get

text: canceled

text: when

text: multiplied

text: by

text: d

text: which

text: cannot

text: be

text: achieved

text: either

text: H

text: A

text: B

text: C

text: D

text: δ

text: by

text: terms

text: with

text: g

text: (they

text: will

text: contain

text: d)

text: or

text: by

text: other

text: heads

text: A

text: C

text: B

text: D

text: A

text: D

text: C

text: B

text: or

text: (they

text: give

text: diﬀerent

text: terms).

text: Therefore

text: we

text: only

text: have

text: to

text: show

text: that

text: heads

text: alone

text: must

text: combine

text: to

text: the

text: A

text: B

text: C

text: D

text: tensor

text: Let

text: us

text: focus

text: ﬁrst

text: focus

text: on

text: a

text: subgroup

text: of

text: these

text: heads

text: (Raδ

text: +Sbδ

text: +Tcδ)(bαcβaγ

text: cαaβbγ).

text: Multiplied

text: by

text: A

text: C

text: B

text: D

text: A

text: B

text: D

text: C

text: d

text: cycles

text: bαcβaγ

text: cαaβbγ

text: must

text: get

text: canceled

text: from

text: gauge

text: invariance,

text: which

text: cannot

text: be

text: done

text: by

text: terms

text: with

text: g

text: (d

text: will

text: J.S.

text: Bell,

text: Physics

text: (Long

text: Island

text: City,

text: N.Y.)

text: P.H.

text: Eberhard,

text: Phys.

text: Rev.

text: A

text: A.

text: Shimony,

text: B.

text: Hensen

text: et

text: al.,

text: Nature

text: J.F.

text: Clauser,

text: M.A.

text: Horne,

text: A.

text: Shimony

text: and

text: R.A.

text: Holt,

text: M.

text: Giustina

text: et

text: al.,

text: Phys.

text: Rev.

text: Lett.

text: Phys.

text: Rev.

text: Lett.

text: L.K.

text: Shalm

text: et

text: al.,

text: Phys.

text: Rev.

text: Lett.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48) ⊆ hence F is a right S-Noetherian R-module. Next, since T = π(N) and Ns K, we have (cid:48) (cid:48) (cid:48) ⊆ π(Ns ) = π(N)s = T s π(K). We know that K is ﬁnitely generated in F , so π(K) is ﬁnitely (cid:48) ⊆ generated R-submodule of D. Thus, T s π(K) which means that T is S-ﬁnite. Since T is an arbitrary R-submodule of D, D is a right S-Noetherian module. Now, we prove that A := R[x; σ] is a right S-Noetherian ring. Let I be right ideal of A and suppose that { ∈ | } ∪ { } J = r R r is a leading coeﬃcient of any polynomial in I 0 . i i ⊆ · · · It is easy to see that J is a right ideal. Since R is right S-Noetherian, Js (a R+a R+ +a R) ∈ ∈ ∈ 1 2· · · n for some{s }S and ai J. So there exist polynomials fi I with fi = ai,nixni + + a0,i. Let d = max n . Assume that T is the set of all polynomials in I with degree less than d. Obviously, i (cid:80) T is a ﬁnitely generated right R-submodule of A. So by the ﬁrst claim, T is right S-Noetherian. ∈ ∈ ≤ ≤ ⊆ · · · Hence there exist t S, g T for 1 i m such that T t (g R + g R + + g R). Let ∈ i ∈ ∈ 1 · · · 2 m − h(x) = z b xi I, so b J which means that b (a R + a R + + a R). Thus hσ z(s) i=1 i z z 1 2 n can be written as follows: − hσ z(s) = v(1) + w(1) + q(1), ∈ · · · ∈ { ∈ | ≤ ≤ − } ∈ where v(1) (f A + f A + + f A), w(1) f A d + 1 deg(f) z 1 and q(1) 1 2 n − − · · · − − T . Continuing in this way and multiplying σ z+1(s), σ z+2(s), , σ 1 d(s) from right side ∈ · · · ∈ respectively, so there exists some v (f A + f A + + f A), w T such that 1 2 n − − · · · − − hσ z(s)σ z+1(s) σ d 1(s) = v + w. − · · · Assume that s = σ z+i and multipling t from right side, then hs s s − t = vt + wt. But ∈ i∈ · · · ⊆ · · · 1 2 z d wt T t, so wt (g R + g R + + g R) (g A + g A + + g A). Hence, 1 2 m 1 2 m · · · ∈ · · · · · · hs s s − t (f A + f A + + f A + g A + + g A). 1 2 z d 1 2 n 1 m ∈ Since s ’s and t are independent from the choice of h I, we have i · · · ⊆ · · · · · · Is s s − t (f A + f A + + f A + g A + + g A). 1 2 z d 1 2 n 1 m · · · ∈ Finally, since s s s − t S, the ideal I is S-ﬁnite and because I was chosen an arbitrary 1 2 z d right ideal of A, hence A is a right S-Noetherian ring. ⇐ ( ) Let I be a right ideal of R. Suppose that { ∈ | } J = f A the leading coeﬃcient of f is in I . (cid:80) ∈ ⊆ Then J is a right ideal of A. Since A is right S-Noetherian, there exists s S such that Js ⊆ · · · K J, where K is a ﬁnitely generated right ideal of A. Suppose that K = (f A+f A+ +f A). ∈ ∈ 1 2 l Let r I, then there exists some f J such that fs = a f . So if r is the leading coeﬃcient ≤ ≤ ∈ · · · ⊆ i i i · · · ⊆ of f , 1 i l, then rs (r R + r R + r R). So Is (r R + r R + + r R). Also, K J, i 1 2 l 1 2 ·l· · ⊆ so each leading coeﬃcient of K is a leading coeﬃcient of J. So (r R + r R + + r R) I and 1 2 l hence I is right S-ﬁnite and R is right S-Noetherian. We have the following generalization of a theorem of D.D. Anderson and Tiberiu Dumitrescu [1, Proposition 9]. ⊆ Corollary 2.5. Let R be a (not necessarily commutative) ring and S R an anti-Archimedean · · · multiplicative set. If R is S-Noetherian then so is the polynomial ring R[X , X , , X ]. 1 2 n 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: hence

text: F

text: is

text: a

text: right

text: S-Noetherian

text: R-module.

text: Next,

text: since

text: T

text: π(N)

text: and

text: Ns

text: K,

text: we

text: have

text: π(Ns

text: π(N)s

text: T

text: s

text: π(K).

text: We

text: know

text: that

text: K

text: is

text: ﬁnitely

text: generated

text: in

text: F

text: so

text: π(K)

text: is

text: ﬁnitely

text: generated

text: R-submodule

text: of

text: D.

text: Thus,

text: T

text: s

text: π(K)

text: which

text: means

text: that

text: T

text: is

text: S-ﬁnite.

text: Since

text: T

text: is

text: an

text: arbitrary

text: R-submodule

text: of

text: D,

text: D

text: is

text: a

text: right

text: S-Noetherian

text: module.

text: Now,

text: we

text: prove

text: that

text: A

text: R[x;

text: σ]

text: is

text: a

text: right

text: S-Noetherian

text: ring.

text: Let

text: I

text: be

text: right

text: ideal

text: of

text: A

text: and

text: suppose

text: that

text: J

text: r

text: R

text: r

text: is

text: a

text: leading

text: coeﬃcient

text: of

text: any

text: polynomial

text: in

text: I

text: i

text: i

text: It

text: is

text: easy

text: to

text: see

text: that

text: J

text: is

text: a

text: right

text: ideal.

text: Since

text: R

text: is

text: right

text: S-Noetherian,

text: Js

text: (a

text: R+a

text: R+

text: +a

text: R)

text: 2·

text: n

text: for

text: some{s

text: }S

text: and

text: ai

text: J.

text: So

text: there

text: exist

text: polynomials

text: fi

text: I

text: with

text: fi

text: ai,nixni

text: Let

text: d

text: max

text: n

text: Assume

text: that

text: T

text: is

text: the

text: set

text: of

text: all

text: polynomials

text: in

text: I

text: with

text: degree

text: less

text: than

text: d.

text: Obviously,

text: i

text: T

text: is

text: a

text: ﬁnitely

text: generated

text: right

text: R-submodule

text: of

text: A.

text: So

text: by

text: the

text: ﬁrst

text: claim,

text: T

text: is

text: right

text: S-Noetherian.

text: Hence

text: there

text: exist

text: t

text: S,

text: g

text: T

text: for

text: i

text: m

text: such

text: that

text: T

text: t

text: (g

text: R

text: g

text: R

text: g

text: R).

text: Let

text: i

text: m

text: h(x)

text: z

text: b

text: xi

text: I,

text: so

text: b

text: J

text: which

text: means

text: that

text: b

text: (a

text: R

text: a

text: R

text: a

text: R).

text: Thus

text: hσ

text: z(s)

text: i

text: z

text: z

text: n

text: can

text: be

text: written

text: as

text: follows:

text: hσ

text: z(s)

text: where

text: (f

text: A

text: f

text: A

text: f

text: A),

text: f

text: A

text: d

text: deg(f)

text: z

text: and

text: n

text: T

text: Continuing

text: in

text: this

text: way

text: and

text: multiplying

text: σ

text: σ

text: σ

text: d(s)

text: from

text: right

text: side

text: respectively,

text: so

text: there

text: exists

text: some

text: v

text: (f

text: A

text: f

text: A

text: f

text: A),

text: w

text: T

text: such

text: that

text: n

text: hσ

text: z(s)σ

text: σ

text: d

text: v

text: w.

text: Assume

text: that

text: s

text: σ

text: z+i

text: and

text: multipling

text: t

text: from

text: right

text: side,

text: then

text: hs

text: s

text: s

text: t

text: vt

text: wt.

text: But

text: i∈

text: z

text: d

text: wt

text: T

text: t,

text: so

text: wt

text: (g

text: R

text: g

text: R

text: g

text: R)

text: (g

text: A

text: g

text: A

text: g

text: A).

text: Hence,

text: m

text: m

text: hs

text: s

text: s

text: t

text: (f

text: A

text: f

text: A

text: f

text: A

text: g

text: A

text: g

text: A).

text: z

text: d

text: n

text: m

text: Since

text: s

text: ’s

text: and

text: t

text: are

text: independent

text: from

text: the

text: choice

text: of

text: h

text: I,

text: we

text: have

text: i

text: Is

text: s

text: s

text: t

text: (f

text: A

text: f

text: A

text: f

text: A

text: g

text: A

text: g

text: A).

text: z

text: d

text: n

text: m

text: Finally,

text: since

text: s

text: s

text: s

text: t

text: S,

text: the

text: ideal

text: I

text: is

text: S-ﬁnite

text: and

text: because

text: I

text: was

text: chosen

text: an

text: arbitrary

text: z

text: d

text: right

text: ideal

text: of

text: A,

text: hence

text: A

text: is

text: a

text: right

text: S-Noetherian

text: ring.

text: Let

text: I

text: be

text: a

text: right

text: ideal

text: of

text: R.

text: Suppose

text: that

text: J

text: f

text: A

text: the

text: leading

text: coeﬃcient

text: of

text: f

text: is

text: in

text: I

text: Then

text: J

text: is

text: a

text: right

text: ideal

text: of

text: A.

text: Since

text: A

text: is

text: right

text: S-Noetherian,

text: there

text: exists

text: s

text: S

text: such

text: that

text: Js

text: K

text: J,

text: where

text: K

text: is

text: a

text: ﬁnitely

text: generated

text: right

text: ideal

text: of

text: A.

text: Suppose

text: that

text: K

text: (f

text: A+f

text: A+

text: +f

text: A).

text: l

text: Let

text: r

text: I,

text: then

text: there

text: exists

text: some

text: f

text: J

text: such

text: that

text: fs

text: a

text: f

text: So

text: if

text: r

text: is

text: the

text: leading

text: coeﬃcient

text: i

text: i

text: i

text: of

text: f

text: i

text: l,

text: then

text: rs

text: (r

text: R

text: r

text: R

text: r

text: R).

text: So

text: Is

text: (r

text: R

text: r

text: R

text: r

text: R).

text: Also,

text: K

text: J,

text: i

text: l

text: ·l·

text: so

text: each

text: leading

text: coeﬃcient

text: of

text: K

text: is

text: a

text: leading

text: coeﬃcient

text: of

text: J.

text: So

text: (r

text: R

text: r

text: R

text: r

text: R)

text: I

text: and

text: l

text: hence

text: I

text: is

text: right

text: S-ﬁnite

text: and

text: R

text: is

text: right

text: S-Noetherian.

text: We

text: have

text: the

text: following

text: generalization

text: of

text: a

text: theorem

text: of

text: D.D.

text: Anderson

text: and

text: Tiberiu

text: Dumitrescu

text: Proposition

text: Corollary

text: Let

text: R

text: be

text: a

text: (not

text: necessarily

text: commutative)

text: ring

text: and

text: S

text: R

text: an

text: anti-Archimedean

text: multiplicative

text: set.

text: If

text: R

text: is

text: S-Noetherian

text: then

text: so

text: is

text: the

text: polynomial

text: ring

text: R[X

text: X

text: X

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ (cid:80) ∈ (cid:80) (cid:80) (cid:48)(cid:48) ∈ some t S and q [N : a]. Since R is duo and S is a denominator set in R, there exists s S (cid:80) (cid:48)(cid:48) j (cid:48)(cid:48) (cid:80) ⊆ · · · such that s at = s ta by Theorem 4.1. Also s(N + aM) (R(n + am ) + + R(n + am )). 1 (cid:48) 1 (cid:48) ∈p p Thus sx = r n + r am . This means that sx = r n + a r m for some r R. Since ∈ (cid:88)i i i i(cid:88)(cid:48) ∈ (cid:88) i (cid:88)i i (cid:88)i i (cid:88) sx, r n N, we have r m [N : a]. So i i i i (cid:48)(cid:48) (cid:48)(cid:48)(cid:80) (cid:80)(cid:48)(cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) s tsx = s t r n + s t ar m = s tr n + s at r m = s tr n + s a c q . i i i i i i i i i i j j (cid:48)(cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) (cid:48) ∈ (cid:48)(cid:48) ∈ · · · (cid:48)(cid:48) So s tsx = s tr n + c s aq for some c R. Hence s tsx (Rn + + Rn + Rs aq + · · · (cid:48)(cid:48) i i(cid:48)(cid:48) ⊆j j · · · j (cid:48)(cid:48) · · · 1(cid:48)(cid:48) ⊆ p 1 + Rs aq ). So s tsN (Rn + + Rn + Rs aq + + Rs aq ) N. Thus N is k 1 p 1 (cid:80) (cid:84) k S-ﬁnite and this contradicts to the fact that N is maximal in F. Therefore P is a prime ideal ⊆ ⊆ of R. Moreover P = [N : M] [N : F ] [N : wM] = [P : w] = P . Hence [N : F ] = P . Let · · · F = (Rf +Rf + +Rf ). Since R is a duo ring, P = [N : Rf ] = [N : f ]. So P = [N : f ] 1 ∈ {2 · · · k } ⊆ i · · · i i for some f f , f , , f . One can show that tN (Rn + Rn + + Rn ) + P M for some ∈ i ∈ 1 2 k 1 2 l t S and n N as above or in similar way as that employed in [1, Proposition 4]. Since P M i ⊆ ⊆ ⊆ ∈ is S-ﬁnite, vP M G P M N for some v S and a ﬁnitely generated submodule G of M. So ⊆ · · · ⊆ (cid:48) (cid:48) · · · (cid:48) ⊆ vtN v(Rn + Rn + Rn ) + vP M (Rn + Rn + Rn ) + G N 1 2 l 1 2 l (cid:48) ∈ for some n N. So N becomes S-ﬁnite which is a contradiction. So M is S-Noetherian. i Lemma 4.3. Let R be a ring with an endomorphism σ. If R[[x; σ]] is a duo ring, then σ is s(cid:80)urjective. (cid:80) (cid:80) ∈ Proof. Suppose that a R. Since R[[x; σ]] is a duo ring we have ax = xf such that f = ∞ ∞ ∞ (cid:54) f xi. So xf = x f xi = σ(f )xi+1. Now, since ax = xf, σ(f ) = 0 for all i = 0 i=0 i i=0 i ∈ i=0 i ∈ i and σ(f ) = a. Thus, for each a R there exists f R such that a = σ(f ). 0 0 0 ⊆ Theorem 4.4. Let R be a ring, S R a σ-anti-Archimedean denominator set (consisting · · · nonzero devisors) and σ , , σ are monomorphisms of R with σ σ = σ σ , for each i, j. · · · 1 ·n· · i j j i Assume that R[[X , , X ; σ , , σ ]] is a duo ring. If R is S-Noetherian, then the ring · · · 1· · · n 1 n R[[X , , X ; σ , , σ ]] is also S-Noetherian. 1 n 1 n Proof. We use the method in [1, Proposition 10] employed by Anderson and Dumitrescu. As S is σ-anti-Archimedean in every ring containing R as a subring, we shall prove the case n = 1, so we assume that T = R[[x; σ]] is duo and σ is an automorphism of R. It is enough to → prove that every prime ideal P of T is S-ﬁnite. Let π : T R the R-algebra homomorphism (cid:80) (cid:80) (cid:48) ∈ sending x to zero and P = π(P ). Since R is S-Noetherian, there exists s S such that (cid:48) ⊆ · · · ∈ ∈ (cid:80) (cid:48) sP (Rg (0)+Rg (0)+ +Rg (0)) for some g P . If x P , then P = (T P +T x). If g (x) = 1 (cid:80) 2 − k ∈ (cid:48) i ⊆ (cid:48) · ·(cid:80)· i ⊆ a xi, then g (x) = xiσ i(a ) (T P + T x). So sP (T P + T x) = (T g + + T g ) P . i i i (cid:80)∈ ∈ (cid:80) 1 k ∈ This means that P is S-ﬁnite. Let x / P and f P . So sf(0) = d g (0) for some d R. (cid:80)− ∈ ∈ ∈ ∈ 0,j j 0,j So xf = sf d g P for some f T . Considering x / P , f P . So sf = d g +xf 1 ∈ 0,j j 1 1 1∈ 1,j j ∈ 2 for some f T . Hence σ(s)sf = σ(s)d g + x d g + x2f . Also f P , since x / P − 2 ∈ 0,j j 1,j j ≥2 2 and sf1 d1,jgj (cid:0)P(cid:89). In th(cid:1)is way(cid:88), one (cid:88)can s(cid:89)ee that for each L 0, L L k L σl(s) f = xi ( σl(s))d g + xL+1f . i,j j L+1 l=0 i=0 j=1 l=i+1 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: some

text: t

text: S

text: and

text: q

text: [N

text: a].

text: Since

text: R

text: is

text: duo

text: and

text: S

text: is

text: a

text: denominator

text: set

text: in

text: R,

text: there

text: exists

text: s

text: S

text: j

text: such

text: that

text: s

text: at

text: s

text: ta

text: by

text: Theorem

text: Also

text: s(N

text: aM)

text: (R(n

text: am

text: R(n

text: am

text: ∈p

text: p

text: Thus

text: sx

text: r

text: n

text: r

text: am

text: This

text: means

text: that

text: sx

text: r

text: n

text: a

text: r

text: m

text: for

text: some

text: r

text: R.

text: Since

text: i

text: i

text: i

text: i

text: i

text: sx,

text: r

text: n

text: N,

text: we

text: have

text: r

text: m

text: [N

text: a].

text: So

text: i

text: i

text: i

text: i

text: s

text: tsx

text: s

text: t

text: r

text: n

text: s

text: t

text: ar

text: m

text: s

text: tr

text: n

text: s

text: at

text: r

text: m

text: s

text: tr

text: n

text: s

text: a

text: c

text: q

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: j

text: j

text: So

text: s

text: tsx

text: s

text: tr

text: n

text: c

text: s

text: aq

text: for

text: some

text: c

text: R.

text: Hence

text: s

text: tsx

text: (Rn

text: Rn

text: Rs

text: aq

text: i

text: ⊆j

text: j

text: j

text: p

text: Rs

text: aq

text: So

text: s

text: tsN

text: (Rn

text: Rn

text: Rs

text: aq

text: Rs

text: aq

text: N.

text: Thus

text: N

text: is

text: k

text: p

text: k

text: S-ﬁnite

text: and

text: this

text: contradicts

text: to

text: the

text: fact

text: that

text: N

text: is

text: maximal

text: in

text: F.

text: Therefore

text: P

text: is

text: a

text: prime

text: ideal

text: of

text: R.

text: Moreover

text: P

text: [N

text: M]

text: [N

text: F

text: [N

text: wM]

text: [P

text: w]

text: P

text: Hence

text: [N

text: F

text: P

text: Let

text: F

text: (Rf

text: +Rf

text: +Rf

text: Since

text: R

text: is

text: a

text: duo

text: ring,

text: P

text: [N

text: Rf

text: [N

text: f

text: So

text: P

text: [N

text: f

text: k

text: i

text: i

text: i

text: for

text: some

text: f

text: f

text: f

text: f

text: One

text: can

text: show

text: that

text: tN

text: (Rn

text: Rn

text: Rn

text: P

text: M

text: for

text: some

text: i

text: k

text: l

text: t

text: S

text: and

text: n

text: N

text: as

text: above

text: or

text: in

text: similar

text: way

text: as

text: that

text: employed

text: in

text: Proposition

text: Since

text: P

text: M

text: i

text: is

text: S-ﬁnite,

text: vP

text: M

text: G

text: P

text: M

text: N

text: for

text: some

text: v

text: S

text: and

text: a

text: ﬁnitely

text: generated

text: submodule

text: G

text: of

text: M.

text: So

text: vtN

text: v(Rn

text: Rn

text: Rn

text: vP

text: M

text: (Rn

text: Rn

text: Rn

text: G

text: N

text: l

text: l

text: for

text: some

text: n

text: N.

text: So

text: N

text: becomes

text: S-ﬁnite

text: which

text: is

text: a

text: contradiction.

text: So

text: M

text: is

text: S-Noetherian.

text: i

text: Lemma

text: Let

text: R

text: be

text: a

text: ring

text: with

text: an

text: endomorphism

text: σ.

text: If

text: R[[x;

text: σ]]

text: is

text: a

text: duo

text: ring,

text: then

text: σ

text: is

text: Proof.

text: Suppose

text: that

text: a

text: R.

text: Since

text: R[[x;

text: σ]]

text: is

text: a

text: duo

text: ring

text: we

text: have

text: ax

text: xf

text: such

text: that

text: f

text: f

text: xi.

text: So

text: xf

text: x

text: f

text: xi

text: σ(f

text: Now,

text: since

text: ax

text: xf,

text: σ(f

text: for

text: all

text: i

text: i

text: i

text: i

text: i

text: and

text: σ(f

text: a.

text: Thus,

text: for

text: each

text: a

text: R

text: there

text: exists

text: f

text: R

text: such

text: that

text: a

text: σ(f

text: Theorem

text: Let

text: R

text: be

text: a

text: ring,

text: S

text: R

text: a

text: σ-anti-Archimedean

text: denominator

text: set

text: (consisting

text: nonzero

text: devisors)

text: and

text: σ

text: σ

text: are

text: monomorphisms

text: of

text: R

text: with

text: σ

text: σ

text: σ

text: σ

text: for

text: each

text: i,

text: j.

text: ·n·

text: i

text: j

text: j

text: i

text: Assume

text: that

text: R[[X

text: X

text: σ

text: σ

text: is

text: a

text: duo

text: ring.

text: If

text: R

text: is

text: S-Noetherian,

text: then

text: the

text: ring

text: 1·

text: n

text: n

text: R[[X

text: X

text: σ

text: σ

text: is

text: also

text: S-Noetherian.

text: n

text: n

text: Proof.

text: We

text: use

text: the

text: method

text: in

text: Proposition

text: employed

text: by

text: Anderson

text: and

text: Dumitrescu.

text: As

text: S

text: is

text: σ-anti-Archimedean

text: in

text: every

text: ring

text: containing

text: R

text: as

text: a

text: subring,

text: we

text: shall

text: prove

text: the

text: case

text: n

text: so

text: we

text: assume

text: that

text: T

text: R[[x;

text: σ]]

text: is

text: duo

text: and

text: σ

text: is

text: an

text: automorphism

text: of

text: R.

text: It

text: is

text: enough

text: to

text: prove

text: that

text: every

text: prime

text: ideal

text: P

text: of

text: T

text: is

text: S-ﬁnite.

text: Let

text: π

text: T

text: R

text: the

text: R-algebra

text: homomorphism

text: sending

text: x

text: to

text: zero

text: and

text: P

text: π(P

text: Since

text: R

text: is

text: S-Noetherian,

text: there

text: exists

text: s

text: S

text: such

text: that

text: sP

text: (Rg

text: +Rg

text: for

text: some

text: g

text: P

text: If

text: x

text: P

text: then

text: P

text: (T

text: P

text: +T

text: x).

text: If

text: g

text: (x)

text: k

text: i

text: i

text: a

text: xi,

text: then

text: g

text: (x)

text: xiσ

text: i(a

text: (T

text: P

text: T

text: x).

text: So

text: sP

text: (T

text: P

text: T

text: x)

text: (T

text: g

text: T

text: g

text: P

text: i

text: i

text: i

text: k

text: This

text: means

text: that

text: P

text: is

text: S-ﬁnite.

text: Let

text: x

text: P

text: and

text: f

text: P

text: So

text: d

text: g

text: for

text: some

text: d

text: R.

text: j

text: So

text: xf

text: sf

text: d

text: g

text: P

text: for

text: some

text: f

text: T

text: Considering

text: x

text: P

text: f

text: P

text: So

text: sf

text: d

text: g

text: +xf

text: j

text: 1∈

text: j

text: for

text: some

text: f

text: T

text: Hence

text: σ(s)sf

text: σ(s)d

text: g

text: x

text: d

text: g

text: Also

text: f

text: P

text: since

text: x

text: P

text: j

text: j

text: ≥2

text: and

text: In

text: one

text: that

text: for

text: each

text: L

text: L

text: L

text: k

text: L

text: σl(s)

text: f

text: xi

text: σl(s))d

text: g

text: i,j

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: These operators remove from f all the bright (opening) or dark (closing) structures where the structuring element B cannot ﬁt. However they also modify the value of pixels when B ﬁt. Thus to avoid these artefacts it has been proposed in [10] to use geodesic opening and { } (i) closing. Then by considering a set γ , i = 1...n, of indexed geodesic openings ,and a set { } R (i) ϕ , i = 1...n, of indexed geodesic closings where typically, the index i is associated to the R size of the structuring element. Then thanks to the granulometry axiomatic [27] we obtain a scale space representation of an image, which allows an image structures decomposition. ∗ Then the Moprhological Proﬁle (MP) of a grey scale image f is deﬁned as 2 n+1 dimension vector:        (1)  γ (f)(x) R  .   .   .    (n) γ (f)(x)  R  MP (x) =  f(x)  (18)    (1)  ϕ (f)(x) R . . . (n) ϕ (f)(x) R To be able to use the MP on hyperspectral images we ﬁrst reduce the dimension of the data thanks to PCA, and then project the data on a d dimensional space which is of smaller dimension than the original space. So a hyperspectral image is represented by d grey scale images, then on each of these images we calculate the MP and we concatenate them. Hence, ∗ ∗ the spatial feature space is of dimension d 2 n + 1. 6.2 Algorithm for the convolutional kernel mean map The formula for the convolutional kernel mean map is deﬁned in equation (7) by : (cid:98) (cid:98) (cid:88) (cid:88) (cid:92) (cid:104) (cid:105) KCN(x , x ) = µ P , µ P i j P( xi) P( xj) (cid:92) (cid:107) (cid:107) (cid:107) (cid:107) KCN(x , x ) = h(x ) . h(x ) i j l1 2 l2 2 ∈P ∈P l1 xi l2 xj e−2β12 (cid:107)xl1−xl2(cid:107)2e−2σ12 .(cid:107)h˜(xl1)−h˜(xl2)(cid:107)2, This equation can be rewritten as:(cid:88) (cid:88) (cid:92) (cid:107) (cid:107) (cid:107) (cid:107) KCN(x , x ) = h(x ) . h(x ) i j l1 2 l2 2 ∈P ∈P l1 xi l2 xj − (cid:107) − (cid:107) e 12 (cid:36)(xl1) (cid:36)(xl2) 2, 14 ##LTLine## ##LTLine## ##LTLine##

text: These

text: operators

text: remove

text: from

text: f

text: all

text: the

text: bright

text: (opening)

text: or

text: dark

text: (closing)

text: structures

text: where

text: the

text: structuring

text: element

text: B

text: cannot

text: ﬁt.

text: However

text: they

text: also

text: modify

text: the

text: value

text: of

text: pixels

text: when

text: B

text: ﬁt.

text: Thus

text: to

text: avoid

text: these

text: artefacts

text: it

text: has

text: been

text: proposed

text: in

text: to

text: use

text: geodesic

text: opening

text: and

text: (i)

text: closing.

text: Then

text: by

text: considering

text: a

text: set

text: γ

text: i

text: of

text: indexed

text: geodesic

text: openings

text: ,and

text: a

text: set

text: R

text: (i)

text: ϕ

text: i

text: of

text: indexed

text: geodesic

text: closings

text: where

text: typically,

text: the

text: index

text: i

text: is

text: associated

text: to

text: the

text: R

text: size

text: of

text: the

text: structuring

text: element.

text: Then

text: thanks

text: to

text: the

text: granulometry

text: axiomatic

text: we

text: obtain

text: a

text: scale

text: space

text: representation

text: of

text: an

text: image,

text: which

text: allows

text: an

text: image

text: structures

text: decomposition.

text: Then

text: the

text: Moprhological

text: Proﬁle

text: (MP)

text: of

text: a

text: grey

text: scale

text: image

text: f

text: is

text: deﬁned

text: as

text: dimension

text: vector:

text: γ

text: (f)(x)

text: R

text: (n)

text: γ

text: (f)(x)

text: R

text: MP

text: (x)

text: f(x)

text: ϕ

text: (f)(x)

text: R

text: (n)

text: ϕ

text: (f)(x)

text: R

text: To

text: be

text: able

text: to

text: use

text: the

text: MP

text: on

text: hyperspectral

text: images

text: we

text: ﬁrst

text: reduce

text: the

text: dimension

text: of

text: the

text: data

text: thanks

text: to

text: PCA,

text: and

text: then

text: project

text: the

text: data

text: on

text: a

text: d

text: dimensional

text: space

text: which

text: is

text: of

text: smaller

text: dimension

text: than

text: the

text: original

text: space.

text: So

text: a

text: hyperspectral

text: image

text: is

text: represented

text: by

text: d

text: grey

text: scale

text: images,

text: then

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