Vacuum Texture: A New Interpretation of Quantum Mechanics and a New Loophole for Bell's Inequality Measurements that preserves Local Realism and Causality
LLA-UR-19-22136 v2
Vacuum Texture: A New Interpretation of Quantum Mechanics and a New Loopholefor Bell’s Inequality Measurements that preserves Local Realism and Causality
Yoko Suzuki ∗ New Mexico Consortium, Los Alamos, NM 87544, USA
Kevin M Mertes † Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: February 21, 2020)We introduce a new interpretation of quantum mechanics by examining the Einstein, Podolskyand Rosen (EPR) paradox and Bell’s inequality experiments under the assumption that the vacuumfluctuation has a locally varying texture (a local variable) for energy levels below the Heisenbergtime-energy uncertainty relation. In this article, selected results from the most reliable Bell’s in-equality experiments will be quantitatively analyzed to show that our interpretation of quantummechanics creates a new loophole in Bell’s inequality, and that the past experimental findings donot contradict our new interpretation. Under the vacuum texture interpretation of quantum me-chanics in a Bell’s inequality experiment, the states of the pair of particles created at the source(e.g. during parametric down conversion) is influenced by an inhomogeneous vacuum texture sentwith the speed of light from the measurement apparatus. We will also show that the resulting pairof particles are not entangled and that the theory of vacuum texture preserves local realism withcomplete causality. This article will also suggest an experiment to definitively confirm the existenceof vacuum texture.
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I. INTRODUCTION
It has been a century since quantum mechanics hastaken the central role in physics. Ever since Einstein,Podolsky and Rosen’s (EPR) [1] question was unexpect-edly answered through experimental measurements [2–4] of Bell’s inequality [5], discussion of the interpreta-tion of quantum mechanics has waned. In the past fewdecades, Bell’s inequality measurements have been per-formed with high precision detection and control. Thesemeasurements have repeatedly confirmed the result andhave seemingly precluded many possible loopholes. Aftera whole century has passed, the vast majority of the sci-entific community seems to completely accept that quan-tum mechanics is a complete theory in the sense thatthere are no hidden variables. Perhaps, the findings fromthe Bell’s inequality measurements are the main reasonswhy hidden variable interpretations are no longer consid-ered as a viable means. A hidden variable theory is stillthe most likely way to model physical nature with localrealism and causality. Here, we would like to revisit theEPR question one more time to see what Bell’s inequal-ity and its past measurements are really telling us andwhether we really have to give up on local realism andcausality.This article will shed light on an assumption that hasbeen carelessly granted; the vacuum was originally as- ∗ [email protected] † [email protected] sumed to be empty, calm and non-interacting. Therehave been some hints, such as zero-point energy, vacuumfluctuation, virtual particles through dark energy, thatthe vacuum should rather be full, stormy and interact-ing. Ultimately, quantum field theory interprets the vac-uum as the completely filled ground states with randomand uncertain fluctuations. Our model of the vacuumassumes it is neither empty nor full, but rather possess-ing a dynamic and deterministic texture (local variable).The details of the theory underlying the interpretationwill be shown elsewhere (ref. [6]).Here, we propose that, instead of vacuum fluctuationsbeing completely stochastic, random and uncertain, theyare locally deterministic and the fluctuations posses aninhomogeneous texture. The texture is created by thepresence of matter (such as an experimental apparatusand the sample itself) and the texture travels with thespeed of light (or very close to it). These low level fluctu-ations are hard to detect since they don’t possess energylarger than what the Heisenberg time-energy uncertaintyprinciple allows. However, vacuum texture can drasti-cally break symmetry during particle creation.In this interpretation of quantum mechanics, the prob-ability distribution is a consequence of an inhomogeneousvacuum texture and the particle is created in one stateat the source and travels in one trajectory. The detectionof the particle does not “collapse” its quantum state assuggested by the Copenhagen interpretation; rather, theplacement of the detector has already broken the sym-metry of the vacuum texture and thus changed the prob-ability of the creation of a particle (and it’s state) along a r X i v : . [ qu a n t - ph ] F e b various trajectories at the source. When the experimen-tal apparatus configuration changes, the vacuum textureis adjusted with (near) luminous speed. In other words,in this interpretation, quantum information is stored inthe vacuum texture. The quantum behavior is the con-sequence of measuring the vacuum texture directly orparticles interacting with vacuum texture. A rigorousformulation of the theory of vacuum texture is beyondthe scope of this article but will be discussed in ref. [6].In this article, we will focus on the new vacuum tex-ture loophole rather than the mechanism behind the in-terpretation. We will suggest an experiment similar tothe one performed by Alain Aspect in 1982 [4]; however,additional data analysis is needed in order to test theexistence of vacuum texture. If the existence of vacuumtexture is verified by an experiment, it will bring a newanswer to the EPR question and will inevitably allow lo-cal variable theories with local realism and causality tobe placed back on the table. II. BELL’S INEQUALITY WITH VACUUMTEXTURE
In a typical Bell’s experiment, a source sends two phys-ical systems (photon’s in this case) to distant observers,Alice and Bob (Fig. 1 (a)). The measurement settingchosen by Alice is a and by Bob is b while the outcomereceived by Alice is α and by Bob is β . For the case ofphoton polarization, a and b correspond to polarizationbeam splitter (PBS) angles. α , β = +1 for detecting aphoton and − E ( a, b ) = (cid:88) α,β = ± αβP ( αβ | ab ) (1)where P ( αβ | ab ) is the probability of the measurementresult for α , β = +1 or − a and b degrees, re-spectively. With standard quantum mechanical analysis,Eq. 1 becomes: E qm ( a, b ) = cos(2( a − b )) , (2)and is shown in Fig. 2 by the red curve.For local hidden variable theories, the probability isexpressed as: P ( αβ | ab ) = (cid:90) Λ dλq ( λ ) P ( α | a, λ ) P ( β | b, λ ) , (3)where, q ( λ ) dλ is the distribution of the hidden vari-able, λ , and Λ represents the manifold of allowed valuesfor λ . For photon polarization, λ is an angle and also (cid:82) π/ − π/ dλq ( λ ) = 1 where q ( λ ) = q ( λ ± π ). AliceBob D ah D av D bh D bv (measure-ment, 𝛼 )(measure-ment, 𝛽 )Pump (a) source s (signal)i (idler) PBS(angle, 𝑎 )PBS(angle, b )AliceBob d f o r A l i c e d f o r B ob Pump vacuum texture traveling with the speed of light (b)
A pair of photons are created.source (measure-ment, 𝛼 )(measure-ment, 𝛽 ) D ah D av D bh D bv PBS(angle, 𝑎 )PBS(angle, b ) FIG. 1. (a) The typical Bell’s inequality measurement set-ting for a pair of photons with a spontaneous-parametric-down-conversion photon source and polarization beam split-ters (PBSs). (b) Under the assumption of vacuum texture,placing an object in a light-like cone that includes the sourceof the creation of a pair of photons effects the measurementand rotational symmetry at the source is broken. In term ofconventional quantum optic, it could be interpreted as thesqueezed vacuum state which no longer has symmetric dis-tribution of uncertainty due to the measurement apparatusbetween the PBS and the source.
In semi-classical analysis, assuming rotational symme-try during the creation of a pair of photons at the source, q ( λ ) = 1 /π , and with Malus’s law, (cid:88) α = ± αP ( α | a, λ ) dλ = cos(2( a − λ )) dλ, (cid:88) β = ± βP ( β | b, λ ) dλ = cos(2( b − λ )) dλ. (4)Using Eq. (1), this yields E sc ( a, b ) = (cos 2( a − b )) / q ( λ ) is independent of a and b : q ( λ | ab ) = q ( λ ). Withinvacuum texture theory, this is no longer true because thevalue of the hidden variable depends on a and b . Thus,the probability of the measurement must be expressedas: P vt ( αβ | ab ) = (cid:90) π/ − π/ dλq ( λ | ab ) P ( α | a, λ ) P ( β | b, λ ) . (5)Bell’s inequality does not hold for Eq. (5) because itrelies on Eq. (3). Eq. (5) can be true even if Bell’s C o rr e l a ti on , E ( a , b ) E qt ( a , b )Semi-classical, E cl ( a , b )Vacuum Texture, E vt ( a , b ) E mclhv ( a , b ) Relative angle, ( a-b ) 𝜋 /8 𝜋 /4 3 𝜋 /8 𝜋 /2- 𝜋 /8- 𝜋 /4-3 𝜋 /8- 𝜋 /2 0 FIG. 2. E qm ( a, b ) is the quantum mechanical correlation func-tion with Alice and Bob’s measurements with settings a and b (red curve). E sc ( a, b ) is for the semi-classical solution us-ing Eq. (3) (blue curve). E vt ( a, b ) is the semi-classical solu-tion with vacuum texture using a local hidden variable in Eq.(5) (black dotted curve, which is identical to the red curve). E mclhv ( a, b ) is maximum classical correlation using (3) whenreplacing Eq. (4) with (cid:80) α = ± αP ( α | a, λ ) = sgn (cos(2( a − λ ))) and (cid:80) β = ± βP ( β | b, λ ) = sgn (cos(2( b − λ ))). inequality is violated; this is the new loop-hole that thetheory of vacuum texture reveals. In Fig. 1 (b), when the pair of photons are created atthe source, the polarization of the created photons areno longer rotationally invariant due to the vacuum tex-ture which was created at the PBS and traveled to thesource. In Fig. 3 and 4, we propose a plausible modelthat can lead to a vacuum texture, which for PBS’s willproduce a vacuum texture with 4-fold symmetry. Whileany specification of a vacuum texture with 4-fold sym-metry is sufficient to determine if it can explain previousBell’s inequality measurement experiments, it is naturalto assume that the PBS angles a , a − π/ b and b − π/ a and b : q ( λ | ab ) dλ = 14 { δ ( λ − a ) + δ ( λ − ( a − π/ δ ( λ − b ) + δ ( λ − ( b − π/ } dλ, (6)Using Eq. (1) and (4)-(6), the correlation function withvacuum texture is calculated as: E vt ( a, b ) = cos(2( a − b )) , (7)Even though we have derived the correlation functionassuming a semi-classical model of photons without anyentanglement, we arrive at an equation that is identicalto quantum mechanical solution that requires entangle-ment: E vt ( a, b ) = E qm ( a, b ). Thus, we have shown thatthis deterministic local hidden variable vacuum texturetheory agrees with the probabilistic aspect of quantummechanical theory. zero-point radiation with rotational invariance (incoming from the detectors)zero-point radiation with 4-fold symmetry in polarization (outward to the source of a pair of photons) 𝝀 q( 𝝀 )=1/4 Alice: a=0° Bob: b=22.5°
HV H V
FIG. 3. A PBS is one of the strongest possible sources ofvacuum texture in the typical Bell’s inequality measurement.It is very plausible that the PBS would transmit a vacuumtexture to the photon source. This vacuum texture does notposses full rotational polarization invariance. Zero-point radi-ation can be thought of as the cause of vacuum texture. In thismodel, without a PBS, zero-point radiation caused by vacuumfluctuations will have a uniform distribution of polarization.However, when the PBS is present, zero-point radiation thatimpinges on the back of the PBS will become polarized, thusbreaking the symmetry. The inset shows q ( λ ) in the polarcoordinate in the 4-fold symmetry with the eight delta func-tions which are expressed in Eq. 6 ( − π/ < λ ≤ π/ | (cid:105) photon state is a quantum statejust as much as a | (cid:105) photon state; so that, the PBS forms asqueezed vacuum state with large uncertainty in the numberof virtual photons in the vacuum. The effects of squeezed | (cid:105) photon states are not negligible especially when the measure-ment intensity is down to | (cid:105) photon states. Thus, tests for the violation of Bell’s inequality can’tbe used to call into question local realism or causalitybecause it is possible to construct a vacuum texture de-scription that yields the exact same quantum mechanicalinequality.
III. FREEDOM OF CHOICE
Bell was highly aware of the possibility that q ( λ | a, b ) (cid:54) = q ( λ ) [5, 8]. Consequently a series measurements wereperformed that added the feature of freedom of choice[4, 9–12]. Two choices/settings are made after/beforethe creation of a pair of photons at the source eitherperiodically or randomly. Alice had a choice between a or a (cid:48) , and Bob had a choice between b or b (cid:48) (see Fig. 5).In the vacuum texture theory with freedom of choice, thefollowing sequence of events occurs:1. Alice/Bob choose a detector setting2. The vacuum texture travels from the detector tothe source in time, d/v vt ≈ d/c
3. The vacuum texture interacts with the source foran interaction time, τ (we assume τ (cid:28) d/c ).4. The photons (with polarization influenced by thevacuum texture/detector setting) travels to each ofthe detectors in time, d/c .The probability function within the vacuum texturetheory with choice becomes: P vt ( αβ | a m b m ( a v b v ))= (cid:90) π/ − π/ dλq ( λ | a v b v ) P ( α | a m , λ ) P ( β | b m , λ ) (8)where a m (which can either be a or a (cid:48) ) and b m (whichcan be either b or b (cid:48) ) are the measurement settings whichwere actually used for the measurement results of α and β , and a v (which can be either a or a (cid:48) ) and b v (which canbe either b or b (cid:48) ) are the measurement settings when theyare in the past lightlike cone of the pair-creation event.In other words, P vt ( αβ | a m b m ( a v b v )) the probability ofmeasuring α and β given the PBS’s were in state a v and b v at the moment the vacuum texture left the PBS’s and the PBS’s were in state a m and b m at the moment thephoton pair emitted from the sample arrive at the PBS’s. (before emission)sourcePump From AliceFrom Bob (after emission) source To AliceTo Bob FIG. 4. A Feynmam diagram depicting a plausible (but notnecessary) model of how a vacuum texture can travel fromthe PBS to the source: The vacuum texture propagates as achain of virtual photons. With this model, we require a polar-ization maintaining interaction between virtual photons. Inother words, the presence of one virtual photon with a partic-ular polarization will induce the existence of another virtualphoton with the same polarization. Without the PBSs, thevacuum texture propagating to the source would be unpolar-ized. However, the presence of the PBSs only allows certainpolarization angles to pass through, creating an inhomoge-neous vacuum texture.
Because of the round-trip time, there are two possiblemeasurement scenarios when a & a (cid:48) and b & b (cid:48) are chang-ing: in-sync and out-of-sync. The in-sync scenario for Al-ice occurs when a m = a v and out-of-sync scenario occurswhen a m (cid:54) = a v , with similar conditions for Bob. In ex-periments conducted to date, Alice and Bob switch theirdetector between two different values either periodicallyor randomly. In either case, the probabilities are accumu-lated over a sufficiently long time interval so that therewill be a fraction of time, f A , when Alice has a m = a v ,in-sync scenario and 1 − f A when she has a m (cid:54) = a v , out-of-sync (with similar definitions for Bob). Here, we assumethat f A is same for a m = a and a m = a (cid:48) ( f B is same for b m = b and b m = b (cid:48) ). This case is especially important for the case for periodic switching which is discussed inthe next section.Under these conditions, the distribution of the hiddenvariable given that the measurement setting are a , b andin the past they could have been a , a (cid:48) , b , b (cid:48) has the fol-lowing form: q fc ( λ | ab ( aba (cid:48) b (cid:48) )) dλ = { f A f B q ( λ | ab )+ f A (1 − f B ) q ( λ | ab (cid:48) )+ (1 − f A ) f B q ( λ | a (cid:48) b ) + (1 − f A )(1 − f B ) q ( λ | a (cid:48) b (cid:48) ) } dλ = { f q ( λ | ab )+(1 − f ) q ( λ | a (cid:48) b (cid:48) )+ f (cid:48) ( q ( λ | ab (cid:48) ) − q ( λ | a (cid:48) b )) } dλ. (9)Where in the last line we used Eq. 6, and have f =( f A + f B ) / f (cid:48) = ( f A − f B ) / With Eq. 8, wefind that the probability for freedom-of-choice when themeasurement settings are a and b under vacuum texturetheory becomes P fcvt ( αβ | ab ( aba (cid:48) b (cid:48) ))= (cid:90) π/ − π/ dλq fc ( λ | ab ( aba (cid:48) b (cid:48) )) P ( α | a, λ ) P ( β | b, λ )= f P vt ( αβ | ab ( ab )) + (1 − f ) P vt ( αβ | ab ( a (cid:48) b (cid:48) ))+ f (cid:48) { P vt ( αβ | ab ( ab (cid:48) )) − P vt ( αβ | ab ( a (cid:48) b )) } . (10)Consequently, the correlation value under freedom-of-choice for actual measurement settings a and b in an ex-periment that has the possible free choices a , b , a (cid:48) , b (cid:48) isexpressed as: E fcvt ( a, b ; a (cid:48) , b ) = (cid:88) α,β = ± αβP fcvt ( αβ | ab ( aba (cid:48) b (cid:48) )) = f E isvt ( a, b ) + (1 − f ) E osvt ( a, b ; a (cid:48) , b (cid:48) ) + f (cid:48) E ubvt ( a, b ; a (cid:48) , b (cid:48) ) . (11)where E isvt ( a, b ) represents the contribution when Aliceand Bob’s detectors are both in sync. E osvt ( a, b ; a (cid:48) , b (cid:48) )represents the contribution when when both their detec-tors are out of sync. The third term could be non-zeroonly when the system is unbalanced: f a (cid:54) = f b ( f (cid:48) (cid:54) = 0).Within vacuum texture theory, for the in-sync term,we revert to the standard correlation value: E isvt ( a, b ) = (cid:88) α,β = ± αβP vt ( αβ | ab ( ab )) = cos(2( a − b )) . (12) With vacuum texture in the case of periodic choice, when ν A = nν B , n=1,2,3... (or ν B = nν A , n=1,2,3...) where ν A and ν B are frequencies of periodic switching for Alice and Bob, f A and f B are no longer independent of each other. In stead, they arecorrelated and depend on their relative phase. In Eq. 9, weassumed that f A and f B are independent. However, using Eq.6, the last line of Eq. 9 still holds for the case of periodic choiceregardless of the relative phase, and the equations are same asthe case of freedom-of-choice. AliceBobPump vacuum texture traveling with the speed of lightA pair of photons are created. PS P S source (measure-ment, 𝛼 )(measure-ment, 𝛽 ) D ah D av D bh D bv d ’ f o r A l i c e d ’ f o r B ob (angle, 𝑎 or a’ )(angle, b or b’ )PBSPBS FIG. 5. Many different schemes have been used to measureBell’s inequality. Here we show a simplified scheme involvinga phase shifter (PS) used as a switch for the measurementsettings of polarization angles between a & a (cid:48) for Alice and b & b (cid:48) for Bob. However, it is important to understand thatregardless of the experimental details there will always exist avacuum texture traveling from the measurement device backto the source which influences the production of the photonsbeing measured. Also, unless otherwise stated, it is assumedthat d (cid:48) ≈ d in Fig. 1 (b). For the out-of-sync term: E osvt ( a, b ; a (cid:48) , b (cid:48) ) = (cid:88) α,β = ± αβP vt ( αβ | ab ( a (cid:48) b (cid:48) ))= 12 { cos(2( a − b ))+cos(2( a + b − a (cid:48) − b (cid:48) )) cos(2( a (cid:48) − b (cid:48) )) } . (13)For the unbalanced term, E ubvt ( a, b ; a (cid:48) , b (cid:48) )= (cid:88) α,β = ± αβ { P vt ( αβ | ab ( ab (cid:48) )) − P vt ( αβ | ab ( a (cid:48) b )) } = 12 { sin(2( a (cid:48) + b (cid:48) − a − b )) sin(2( a (cid:48) − b (cid:48) )) } . (14)The experiments often measure Bell’s S value for a =0, b = π/ a (cid:48) = π/ b (cid:48) = 3 π/ | S | ≤
2. It is easy to showthat: S fcvt (0 , π/ , π/ , π/
8) = 2 √ f, (15)where S fcvt ( a, b, a (cid:48) , b (cid:48) ) = | E fcvt ( a, b ; a (cid:48) , b (cid:48) ) − E fcvt ( a, b (cid:48) ; a (cid:48) , b )+ E fcvt ( a (cid:48) , b ; a, b (cid:48) ) + E fcvt ( a (cid:48) , b (cid:48) ; a, b ) | . (16)From the above analysis, it is clear that depending uponthe experimental details of the particular freedom-of-choice experiment, under the vacuum texture theory,Bell’s S value can range anywhere from the quantumvalue of 2 √
2, to the semi-classical value of √
2, and allthe way to 0. If the choice is completely random (like most of theexperiments were intended [9–12]), f = 1 / f (cid:48) = 0so that Eq. 11 simplifies to: E rcvt ( a, b ; a (cid:48) , b (cid:48) ) = E isvt ( a, b ) + E osvt ( a, b ; a (cid:48) , b (cid:48) )2 . (17)The Bell’s S value for random choice : S rcvt (0 , π/ , π/ , π/
8) = √ ≈ . < . (18)Interestingly, this equals the value expected from a semi-classical model, S sc = √
2. This is not consistent withmany freedom-of-choice measurements [9–12]. However,those measurements, involving single-photon intensitiesextended to many kilometers, suffer from very poor sam-pling rates. Which in turn decreases our ability to makeany definitive claims. For example, a fair sampling as-sumption is applied to the data. Shown in Fig. 2, thesemi-classical result (blue curve) which can be made toresemble the quantum one (red curve) simply by scalingthe data as is needed to impose fair sampling.Moreover, in the vacuum texture model, as the signal-to-noise ratio decreases, the signals depend more on thevacuum texture itself which is created by the various ex-perimental apparatus near the detector and depends lesson the consequence of the incident excitation (i.e. the ac-tion of the pump beam on the source) as many low-signalexperiments depend on timed-window detection ratherthan signal-triggered detection. In any cases, randomchoice is not the best measurement setting to prove ordisprove the existence of the vacuum texture. A moresuitable measurement is suggested in the next section.
IV. SUGGESTED EXPERIMENT
In 1982, Aspect et al. performed the most cited Bell’sinequality measurement [4]. This was the first experi-ment to successfully introduce choice . The PBS anglesof a and a (cid:48) for Alice, and b and b (cid:48) for Bob were switchedperiodically at a frequency around 50 MHz by introduc-ing a phase delay in each path via an electro-optical phasemodulator [13, 14].In the 1982 Aspect experiment, a different form ofBell’s inequality, S (cid:48) was used: S (cid:48) = N ( a, b ) N ( ∞ , ∞ ) − N ( a, b (cid:48) ) N ( ∞ , ∞ (cid:48) ) + N ( a (cid:48) , b ) N ( ∞ (cid:48) , ∞ ) + N ( a (cid:48) , b (cid:48) ) N ( ∞ (cid:48) , ∞ (cid:48) ) − N ( a (cid:48) , ∞ ) N ( ∞ (cid:48) , ∞ ) − N ( ∞ , b ) N ( ∞ , ∞ ) , (19)where N ( ∞ , ∞ ) is without a PBS, and N ( a, b ) /N ( ∞ , ∞ ) ≡ n ( a, b ) = P ( α = β = 1 | ab ) isthe probability of a photon being detected with asingle detector measurement for both Alice and Bobwith the PBS setting angle a and b and with Malus’slaw, P ( α = 1 | a, λ ) dλ = cos ( a − λ ) dλ . Therefore, intheory, N ( a, ∞ ) /N ( ∞ , ∞ ) = N ( ∞ , a ) /N ( ∞ , ∞ ) = 1 / − (cid:54) S (cid:48) (cid:54) Bob
The optical switch setting at the time of vacuum texture departing from the switch at the time t = t . Alice
The optical switch setting at t = t +2 d/c . a a’b b’ in sync out of syncin syncout of sync FIG. 6. In the Aspect experiment, f A = 0 . f B = 0 . f = 0 .
90 and f (cid:48) = 0 . With a , b , a (cid:48) and b (cid:48) of 0, π/ π/ π/
8, respec-tively, Eq. 19 simplifies to S (cid:48) qm = 0 .
207 for the standardquantum mechanical interpretation, since n qm ( a, b ) =cos ( a − b ) /
2. With vacuum texture without choice wealso have, S (cid:48) vt = 0 .
207 since n vt ( a, b ) = cos ( a − b ) /
2. Forthe semi-classical case, S (cid:48) sc = − .
146 where n sc ( a, b ) = (2 + cos(2( a − b ))).Similar to Eq. 11, under vacuum texture with freedom-of-choice, we have n fcvt ( a, b ; a (cid:48) , b (cid:48) ) = f n isvt ( a, b ) + (1 − f ) n osvt ( a, b ; a (cid:48) , b (cid:48) )+ f (cid:48) n ubvt ( a, b ; a (cid:48) , b (cid:48) ) (20)where n isvt ( a, b ) = n qm ( a, b ) = cos ( a − b ) / n osvt ( a, b ; a (cid:48) , b (cid:48) ) = P vt ( α = β = 1 | ab ( a (cid:48) b (cid:48) )) = { a − b )) + cos(2( a + b − a (cid:48) − b (cid:48) )) cos(2( a (cid:48) − b (cid:48) )) } is the out-of-sync component, and n ubvt ( a, b ; a (cid:48) , b (cid:48) ) = P vt ( α = β = 1 | ab ( ab (cid:48) )) − P vt ( α = β = 1 | ab ( a (cid:48) b )) = { sin(2( a (cid:48) + b (cid:48) − a − b )) sin(2( a (cid:48) − b (cid:48) )) } is the unbalancedcomponent.Therefore, with a , b , a (cid:48) and b (cid:48) of 0, π/ π/ π/ S (cid:48) fcvt (0 , π/ , π/ , π/
8) = −
12 + f √ S (cid:48) fcvt ( a, b, a (cid:48) , b (cid:48) ) = n fcvt ( a, b ( a (cid:48) , b (cid:48) )) − n fcvt ( a, b (cid:48) ( a (cid:48) , b ))+ n fcvt ( a (cid:48) , b ( a, b (cid:48) )) + n fcvt ( a (cid:48) , b (cid:48) ( a, b )) − − . (22)For the periodic switching case, we can obtain f A (and f B ) from the switching frequencies ν A (and ν B ). For Al-ice, f A has the maximum condition if an integral multipleof the wave length would fit in the light path, 2 d whichis the round trip distance between PBS and the source.2 d = λ A n → f A = 12 d = λ A ( n + 12 ) → f A = 0 (23) where λ A = c/ν A and n = 0 , , , ... If we assume theswitching to be a perfect square wave, for Alice, f A = 1 π arccos (cos { π ( 2 dc ν A −
12 ) } ) , (24)with similar definitions for Bob. S v a l u e Semi- classical value Quantum value Bell’s inequality
FIG. 7. The expected S (cid:48) value with vacuum texture is plottedin a function of switching frequency. The red dot is at thefrequency which was used for Alice, and the blue dot is at thefrequency which was used for Bob in [4]. The measured S (cid:48) isthe average of the settings of Alice and Bob (see Eq. 21 and24). Expected S value in Eq. 15 under the same conditionsis also shown in orange. The green dotted line represents thequantum value. The purple dotted line represents the semi-classical value. The shaded region shows Bell’s inequality. In the 1982 Aspect experiment, the round trip timefor light between the switch and the pair of photonsource was 2 d/c = 43 ns for Alice and Bob. The opti-cal switching frequency for Alice was 46.2 MHz and thenthe switching period was 43.3 ns. The optical switchingfrequency for Bob was 48.4 MHz and then the switchingperiod was 41.3 ns [4, 13, 14]. In 1989, Zeilinger pointedthat “there was a numerical coincidence between photonflight time and switching frequency [15].” In this mea-surement, the average of Alice and Bob had 90 % of thetime in phase, f = 0 . S (cid:48) fcvt = 0 . S (cid:48) by a small amount. Themeasured value in ref. [4] was S (cid:48) measured = 0 . ± . S (cid:48) = 0 .
207 ( S = 2 .
83) arequantum effects in which, in this new interpretation, theincident particles are in-sync with vacuum texture. Atthe minima down to S (cid:48) = − . S = 0), which cor-responds to the completely out-of-sync situation, evenclassical correlation is diminished. If the frequency set-ting are random and averaged, the in-sync and out-of-sync components would average out to the semi-classicalbehavior, S (cid:48) sc = − .
146 ( S = 1 . d (cid:48) (see Fig. 5) or d without a phase shifter (seeFig. 1 (b)) for Alice and Bob are equal. When they aredifferent, we might be able to observe the dependence ofthe effects of the vacuum texture on the distance. Forexample, the switch for Alice can be removed, and Al-ice’s setting can be fixed at a or a (cid:48) with distance d a fora set of measurements while Bob can have the same pe-riodical switching between b and b (cid:48) with distance d (cid:48) b . At d a (cid:29) d (cid:48) b , the frequency sweep for Bob should show themaxima and minima with the same periodicity in Fig.7 in S (cid:48) . Under these circumstances the delta functionsappearing in Eq. 6 will have unequal weights for Aliceand Bob. The maxima and minima in S (cid:48) should disap-pear in the limit where d a (cid:28) d (cid:48) b , and the value should beconstant. Measurements taken when d a ≈ d (cid:48) b would bevery interesting because it would enable us to determinethe amplitudes of the weight functions as a function ofdistance appearing in Eq. 6. V. DISCUSSION
In our interpretation, we have shown that quantumnonlocality such as superposition and entanglement arenot needed to describe Bell’s inequality measurementsand perhaps other measurements as well. Vacuum tex-ture (a local variable) may also easily explain away theso-called observations of quantum-to-classical transitionsuch as dephasing. As we showed, the interaction withvacuum texture (a local variable) is in-sync or out-of-sync and results in maxima or minima of expectationvalues of measured quantities. If the settings of an ex-perimental apparatus, during the experiment, is unstableor in an uncontrollable environment due to effects suchas thermal fluctuations, the effects of the vacuum texture(a local variable) on the expectation values will becomewashed out. Consequently, the in-sync and out-of-sync effects are averaged out to the classical value within thedephasing time, T . In the classical world, the settingsof the environment would change due to thermal fluc-tuation as the vacuum texture travels between masses.This could be the reason that quantum effects are of-ten observed only at low temperatures in relatively smallsystems.In Fig. 7, there is no frequency dependence in theconventional interpretation of quantum mechanise (greendotted line). Our vacuum texture interpretation with alocal variable with complete causality reveals more de-tailed information (black solid lines) with maxima andminima. The semi-classical value is at the averaged valuewhen the maxima and minima structure is washed out.The most cited Bell’s inequality measurement of 1982Aspect’s measurement [4] has never been reproduced byanyone. The non-constant S values at different frequen-cies might startle experimentalists. However, this couldbe the most important information in physics over thepast century. VI. CONCLUSION
We have proposed that the existence of vacuum tex-ture would explain Bell’s inequality measurements withlocal realism and causality. An experiment, which wascapable with 1982 technology, is suggested for definiteproof/disproof of vacuum texture. If the existence of thevacuum texture is proven by experiment, we will finallyhave the definitive answer to the EPR question, and alocal variable theory should be back on the table as apossible interpretation of quantum mechanics.
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