Does the New Quantization Method Redefine Time Symmetry in Quantum Mechanics?

[Dickfore]

From what I was able to understand by skimming through the text, the author is led by the discreteness of the Matsubara frequencies for the Fourier transform of the fermionic and bosonic Green's functions in the imaginary time formalism. It is true that this is a consequence of the periodicity in imaginary time, however, this periodicity is purely formal due to the similar form of the time evolution operator and the canonical statistical operator.

Eq.(2) seems to impose the periodic boundary conditions with respect to time in this way. However, I suspect that this actually ruins the Lorentz invariance. Namely, there is only one inertial frame where this condition can be stated in the given manner. In all other frames, these space-time points are not synchronous.

 

[naturale]

From what I was able to understand by skimming through the text, the author is led by the discreteness of the Matsubara frequencies for the Fourier transform of the fermionic and bosonic Green's functions in the imaginary time formalism. It is true that this is a consequence of the periodicity in imaginary time, however, this periodicity is purely formal due to the similar form of the time evolution operator and the canonical statistical operator.

In physics the boundary between mathematical tricks and physical reality is very tiny. But, w.r.t Matsubara, in that theory it is the real (minkowskian) time to be periodic and not the imaginary (euclidean) one. If you want you can think to the assumption of intrinsic periodic fields as a mathematical trick.

Eq.(2) seems to impose the periodic boundary conditions with respect to time in this way. However, I suspect that this actually ruins the Lorentz invariance. Namely, there is only one inertial frame where this condition can be stated in the given manner. In all other frames, these space-time points are not synchronous.

This is for sure the first thing that comes to mind. The author takes particular care to show that the consistence with Lorentz invariance and respect relativistic causality, see par.2.4 and fig.3.
It is important to note that together with time periodicity T_t, induced spatial periodicities \vec\lambda_x must be considered. The space-time periodicities are T^\mu = \{ c T_t , \vec\lambda_x \} . They are space-time intervals and therefore they transform according to Lorentz.

From the abstract: "The theory, whose fundamental set-up is presented in this paper, turns out to be consistent with special relativity and in particular respects causality." This can be seen in various ways.
1) Basically, periodic boundary conditions minimize the relativistic action at the boundaries. In this way the relativistic symmetries of the action are preserved.
2) The space-time periodicities are not static, of course! They are the usual de Broglie periodicities, T^\mu = h / p_\mu. They are dynamical by construction. This is similar to the fact that the euclidean periodicity of in field theory at finite temperature is "thermodynamical": when temperature changes, euclidean periodicity changes. In that theory when energy-momentum p_\mu changes through interaction or Lorentz transformations, the space-time periodicities T^\mu change as well. Since that field theory is based upon relativistic waves propagating with usual green function, the four-momentum changes according to the retarded potentials. Therefore the periodicities changes in agreement with relativistic causality.
3) The space-time periodicities T^\mu = h / p_\mu form a controvariant four vector in order to leave the phase of the field invariant under periodic translations. By definition space-time periodicity T^\mu means: exp[- i x_\mu p^\mu] =exp[-i (x_\mu + T_\mu) p^\mu].
4) At a more fundamental and conceptual level you have an highlighting justification. From par.3.2: "Paraphrasing the Newton’s law of inertia [p_\mu = const] and the de Broglie hypothesis of periodicity [T^\mu = h / p_\mu], we assume that every isolated elementary system (every free elementary field) has persistent and constant [space-]time periodicity (as long as it doesn’t interact)" . "But, as much as the Newton’s law of inertia doesn’t imply that every point particle goes in a straight line, our assumption of periodicity does not mean that the physical world should appear to be periodic. In fact there is not a single static periodicity which would serve as privileged reference. On the contrary elementary systems (that we represent as fields) at different energies have different periodicities.21" "Furthermore, through interactions the elementary systems pass from a periodic regime to another periodic regime, forming in general ergodic and even more chaotic evolutions."

Nice and easy!

The amazing point is that the assumption of periodicity provides a quantization condition for relativistic fields and the resulting theory matches ordinary quantum field theory!

P.S: There are problems with the mathematical environment of the Forum when you "go advanced" and see "preview".

 

[Dickfore]

This is for sure the first thing that comes to mind. The author takes particular care to show that the consistence with Lorentz invariance and respect relativistic causality, see par.2.4 and fig.3.

It is interesting you say that the author pays particular attention to this issue (in paragraph 1.3, not 2.4), but he only devotes barely a page of a 30 page booklet. In comparison, the Introduction is four pages long.

I stand on my assertion that eqn. (3) is not Lorentz invariant.

Also, what does "massive field rest frame" actually mean?

 

[zonde]

I stand on my assertion that eqn. (3) is not Lorentz invariant.

This assertion seems valid.
Under Lorentz transformation spacetime interval is invariant when it has the same "slope" as speed of light.
So if we rewrite eqn.(3) like that:
\Phi(x,t)\equiv\Phi(x+\lambda ,t+2\pi R_t)
or like that:
\Phi(x-x_0,t)\equiv\Phi(x_0-x+\lambda ,t+2\pi R_t)
we would have Lorentz invariance.
At least so it seems to me.

 

[naturale]

It is interesting you say that the author pays particular attention to this issue (in paragraph 1.3, not 2.4), but he only devotes barely a page of a 30 page booklet. In comparison, the Introduction is four pages long.

No. The issue is discussed throughout the whole paper. Par.1.3 (Lorentz transformations and covariance) and par.1.4 (Retarded potential and causality) are focalized essentially on that. But also par.1.1 and par.1.2 are devoted to build a relativistic field theory. Par.3.2 shows the conceptual meaning of the assumption of intrinsic periodicities.

I stand on my assertion that eqn. (3) is not Lorentz invariant.

Eqn.3 or eqn.2 are not covariant, right! but read few line above eqn.2: "For simplicity in this preliminary discussion we will concern only with time dimension boundaries." and few lines above par.1.1 : "Provided analogous periodic conditions along the spatial and, for massive fields along the proper time dimensions, such as to guarantee covariance, we shall see that this theory of periodic fields is consistent with special relativity." After that the author shows that time periodicity induces spatial periodicity "such as to guarantee covariance", see eqn.10, and for massive fields, proper time periodicity eqn.20 or eqn.16. "Induced" means that if you assume a relativistic wave with a give time period, of curse you automatically have a spatial periodicity.

Also, what does "massive field rest frame" actually mean?

[/QUOTE]

I can't find this expression in the arXiv version. Probably it means "the rest frame of the massive field". Intrinsic time periodicity for a massive field implies intrinsic periodicity of the proper time, see fig.(2). For massless fields the proper time periodicity is infinite, see.fig.1 (the photon can't be observed in its rest frame). The relation is
T_\tau^{-2} = T_t^{-2} - c^2 \lambda_x^{-2}, where T_\tau = proper time periodicity, T_t time periodicity, \lambda_x spatial periodicity. If you multiply this for "h" you obtain M^2 c^4 = E^2 - p^2 c^2, In fact h/T^2_\tau = M^2 c^4 eqn.16, h/T^2_t = E^2 eq.(8), h^2 c^2 / \lambda_x^2 = p^2 c^2 eqn.10 (T= 2 \pi R).

 

[naturale]

Right Zonde! That is the periodicity that you must consider (if you meant x -x_0 in both sides of the second equation)! The author says "there is a single periodicity which is induced to the other dimensions". If you start from time periodicity and you impose relativistic kinematics you necessarily obtain spatial periodicity, and proper time periodicity for massive objects. Defining T^\mu = \{c T_t, \vec \lambda_x \} you can write it in covariant way
\Phi(x^\mu_0)\equiv\Phi(x^\mu_0+T^\mu).

 

[zonde]

Enough to say that the assumption of de Broglie clocks allows an EPR gedanken experiment without involving hidden variables in order to fulfill Bell or CHSH inequalities which can be tested by using polarized photons. From a mathematical point of view, the quantum states, superposition of pure states, are perfectly reproduced by quantizing a field by periodic boundary conditions. But don't ask me how this can be interpret in words. I have to think about that.

If you want to defend your statement you have to provide some arguments and should be ready to argue against counter arguments.

Well, about this point I go back to the Bell theorem and I remember you that theories with local hidden variable doesn't make quantum mechanics. More. from, http://arxiv.org/abs/0904.1655, State-independent experimental test of quantum contextuality
G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, C. F. Roos
(Submitted on 10 Apr 2009 (v1), last revised 5 May 2009 (this version, v2))
Journal reference: Nature 460, 494 (2009)
DOI: 10.1038/nature08172 ,
it seems that generic (local or not) hidden variables have problems.

Well you have to take in account something about no-go theorems and in particular their experimental verification.
Traditional theorems make positive statements and we know very well how to verify them experimentally. The question is can we do it the same way with no-go theorems.

Now imagine that we have space where we encircle some subspace and make a statement that some X is withing that enclosed area (A). To test this statement we have additional restriction that we can't use exactly the same subspace for testing purposes. We can only use bigger subspace (B) that encloses A or smaller subspace (C) that is enclosed within A.
Obviously for testing purposes we should use subspace C because if we would find out that X is within C it will be within A as well.

But now consider statement where we say that X is outside A.
Just as obvious that in this case for testing purposes we should use subspace B as in this case if we would find out that X is not within B it will not be within A as well.

What I want to say with this example is that requirement to close detection and locality loopholes in the same experiment is not just folly of proponents of local realism but quite valid requirement.

Of course it is useful to analyze even non-conclusive experiments but they can not be considered as a proof of violation of local realism.

I do not agree completely with this. In fact the most peculiar quantum behaviors arise in the relativistic limit. Electromagnetic field is always relativistic and it is quantized at high frequencies. The quantum limit of massive particle is the relativistic limit. This is clear from the Feynman path integral. To understand the deepest nature of quantum physics one should investigate the relativistic limit, and more generically, the relativistic meaning of time. If you are interested on the non relativistic quantum mechanics you may read par.2.4 of 0903.3680v5 and the appendixes of 0903.3680v4. There are showed simple applications to Schrödinger problems.

I do not quite see your point. Entanglement is explored mainly within non relativistic setups i.e. there is laboratory frame and everything is analyzed within that frame. And your statement that I am questioning is about entanglement.

 

[twofish-quant]

But also the periodic fields is a superposition of pure states, it can written in terms of Hilbert eigenstates! See eq.(31) for the definition of a generic Hilbert state and eq.(41) for the definition of the expectation value of an observable.

eq.(31) and eq.(41) will work for a single particle, but the problem with Bell's inequality comes up when you have a field consisting of multiple particles where you have correlations between the that you can't get assuming that the particles are independent probabilistically.

There you can find all the answers you need.

I did. It's a *very* interesting paper, and it's *possible* that you can structure the fields in a way that you get back quantum mechanics, but that hasn't been done yet, and eq. (31) to eq.(41) is insufficient to convince me that they've resolved the Bell's theorem issues.

 

[twofish-quant]

Right Zonde! That is the periodicity that you must consider (if you meant x -x_0 in both sides of the second equation)! The author says "there is a single periodicity which is induced to the other dimensions". If you start from time periodicity and you impose relativistic kinematics you necessarily obtain spatial periodicity, and proper time periodicity for massive objects. Defining T^\mu = \{c T_t, \vec \lambda_x \} you can write it in covariant way
\Phi(x^\mu_0)\equiv\Phi(x^\mu_0+T^\mu).

I think there is a way of doing an experimental test of this. You use the "strobe light" effect. Suppose you have something that is periodic at period n where n is much smaller than what you can detect. If you perform an experiment that samples a particle at 100000.1 n, then you ought to see a systematic strobe light effect.

I'm still not sure how you get Bell's inequality to work. The problem is that the field that you are looking at may be periodic, but you can control the experiment so that time (and hence the state when the particle his the polarizer) is random.

 

[zonde]

if you meant x -x_0 in both sides of the second equation

No I meant what I wrote. Otherwise particle would move at speed of light but that's the first equation. You can apply second equation two times and you get:
\Phi(x-x_0,t)\equiv\Phi(x_0-x+\lambda ,t+2\pi R_t)\equiv\Phi(x-x_0,t+4\pi R_t)
So you have particle at rest with "trembling" along x direction.

 

[naturale]

I try to adapt the ERP experiment to our case.

Let's suppose a coin with arrow up and arrow down on the two sides respectivelly. This coin rolls with period 10^-21 s (~ ZHz). This rolling is not given by any hidden variable, but it is given by the intrinsic periodicity T associated to the mass of the electron at rest which is T~10^-21 s . We must suppose that the coin rolls in the dark and suddenly a stroboscopic light is turned on for an infinitesimal instant. This corresponds to the emission of the polarized photons from the electron. Then Alice and Bob see up and down or down and up, but if their resolution in time is poor they can predict the outcomes only statistically, and the probabilities are 50% and 50%. As Alice see the up arrow, she automatically knows that Bob will see the down arrow or vice versa. Since no hidden variables are involved, we are not in the hypothesis of the Bell theorem and the model can in principle satisfy the Bell inequality.

To resolve the deterministic mechanics of such a coin you need to measure time with a clock whose period is smaller than 10^-21 s (the emission of the polarized photon necessarily involved an electron). To have a precision of T*100000.1 you need a clock with periodicity faster than 10^-22 s in order to be sure to turn on the stroboscopic light at the right instant. The cesium atomic clock has a period of 10^-10 s. The difference between the cesium atomic clock and the de broglie clock of an electron is similar to the difference between a year and the age of the universe! I fear that it is impossible to achieve this time resolution. But if you are able to build such a clock you could in principle predict the outcomes.

 

[naturale]

No I meant what I wrote. Otherwise particle would move at speed of light but that's the first equation. You can apply second equation two times and you get:
\Phi(x-x_0,t)\equiv\Phi(x_0-x+\lambda ,t+2\pi R_t)\equiv\Phi(x-x_0,t+4\pi R_t)
So you have particle at rest with "trembling" along x direction.

No! You have not to considering a particle moving between 0 and T_t, but a string vibrating in space-time with intrinsic space-time periodicities or in compact space-time dimensions. Think to a guitar string, for instance or to the quantization of a "particle" in an infinite well.

The correct condition that you must impose (if you suppose no interactions) is
\Phi(x,t)\equiv\Phi(x+\lambda ,t+T_t)\equiv\Phi(x+n \lambda,t+n' T_t) with n and n' integer numbers (if in a certain space-time point the string interact the periodicities change).

 

[zonde]

I try to adapt the ERP experiment to our case.

Let's suppose a coin with arrow up and arrow down on the two sides. This coin rolls with period 10^-21 s (~ ZHz). This rolling is not given by any hidden variable, but it is given by the intrinsic periodicity T associated to the mass of the electron at rest which is T~10^-21 s . We must suppose that the coin rolls in the dark and suddenly a stroboscopic light is turned on for an infinitesimal instant. This corresponds to the emission of the polarized photons from the electron. Then Alice and Bob see up and down or down and up, but if their resolution in time is poor they can predict the outcomes only statistically, and the probabilities are 50% and 50%. As Alice see the up arrow, she automatically knows that Bob will see the down arrow or vice versa. Since no hidden variables are involved, we are not in the hypothesis of the Bell theorem and the model can in principle satisfy the Bell inequality.

Photons are produced at place S and time T_0. They are measured at place A after time T_1 and at place B after time T_2. You measure photons after two different time periods.
If we take your dice rolling in the dark then we make observations at two different times. There is no reason so far why they should produce the same result.

 

[naturale]

Photons are produced at place S and time T_0. They are measured at place A after time T_1 and at place B after time T_2. You measure photons after two different time periods.
If we take your dice rolling in the dark then we make observations at two different times. There is no reason so far why they should produce the same result.

Use a little bit of imagination. I meant that the stroboscopic light is tuned on and shows the coin at place S and time T_0 for an infinitesimal instant. Alice looks at a face of the coin and B looks at other side. Alice in A and at time T_1 (that is after a time delay T_1 - T_0 depending on how far she is from the coin) sees the face (the photon emitted by the that face of the coin at T_0 and S) with an up arrow (down arrow) whereas Bob in B sees the other face (the photon emitted by the other face at T_0 and S) with the down arrow (up arrow) at time T_2. Alice knows at T_1 what Bob will see at time T_2. It is clear now?

 

[Dickfore]

No. The issue is discussed throughout the whole paper. Par.1.3 (Lorentz transformations and covariance) and par.1.4 (Retarded potential and causality) are focalized essentially on that. But also par.1.1 and par.1.2 are devoted to build a relativistic field theory. Par.3.2 shows the conceptual meaning of the assumption of intrinsic periodicities. Eqn.3 or eqn.2 are not covariant, right! but read few line above eqn.2: "For simplicity in this preliminary discussion we will concern only with time dimension boundaries." and few lines above par.1.1 : "Provided analogous periodic conditions along the spatial and, for massive fields along the proper time dimensions, such as to guarantee covariance, we shall see that this theory of periodic fields is consistent with special relativity." After that the author shows that time periodicity induces spatial periodicity "such as to guarantee covariance", see eqn.10, and for massive fields, proper time periodicity eqn.20 or eqn.16. "Induced" means that if you assume a relativistic wave with a give time period, of curse you automatically have a spatial periodicity.

That may be so, but now I find an even deeper problem. From the first paragraph on p.9:

We approach the theory as a Kaluza-Klein theory for a massless five-dimensional Klein-
Gordon field with periodic extra-dimension s and periodic real time. In fact the resulting
five-dimensional metric is dS^{2} = c^{2} dt^{2} - d\mathbf{x}^{2} - ds^{2} = 0 , so that assuming s = c \tau we recover the usual four-dimensional Minkowskian metric eq.(1). For this reason we will say
that the proper time acts as a “virtual” extra-dimension4 whose length is therefore fixed
by the time periodicity in the rest frame.

Now, there are 3 IMPORTANT, but IMPLICIT assumptions made here:

1. That space time is actually five dimensional. This would mean that the general Lorentz transformations belong to the group SO(1, 3 + 1) and this one has different irreducible representations from the ordinary SO(1, 3) group. Since the spin of the particles is exactly dependent on the representations of this group, the author could have at least given an effort to clarify this issue a bit more, since the whole subsequent derivation hinges on this fact;
2. That they only consider null geodesics dS^{2} = 0. I don't see any a priori reason for this and it is a huge constraint by itself. Saying the field is massless 5d is not an answer. The existence of a space-time metric between any two points should have nothing to do with the matter fields;
3. That one of the dimensions is compactified. In doing this, you essentially break the Lorentz invariance of the theory.

I can't find this expression in the arXiv version. Probably it means "the rest frame of the massive field". Intrinsic time periodicity for a massive field implies intrinsic periodicity of the proper time, see fig.(2). For massless fields the proper time periodicity is infinite, see.fig.1 (the photon can't be observed in its rest frame). The relation is
T_\tau^{-2} = T_t^{-2} - c^2 \lambda_x^{-2}, where T_\tau = proper time periodicity, T_t time periodicity, \lambda_x spatial periodicity. If you multiply this for "h" you obtain M^2 c^4 = E^2 - p^2 c^2, In fact h/T^2_\tau = M^2 c^4 eqn.16, h/T^2_t = E^2 eq.(8), h^2 c^2 / \lambda_x^2 = p^2 c^2 eqn.10 (T= 2 \pi R).

I meant specifically the continuation of the last paragraph from p.8 on p.9 and the footnote 3. A quote:

The key assumption for a massive relativistic field is that it is possible to choose a reference
system (the rest frame) where the real time and the proper time can be identified.3
Therefore, for massive fields, we must consider that the compactification of the real time
induces a compactification of the proper time, as well.

The basic characteristic of fields is that they can support excitations that propagate and carry certain momentum and energy. However, the number of excitations is essentially unlimited (except by the Pauli Exclusion Principle for fermionic fields). In this way, we might as well have two particle excitations moving in opposite directions. What is the rest frame of the field in this case?

 

[naturale]

That may be so, but now I find an even deeper problem. From the first paragraph on p.9:
Now, there are 3 IMPORTANT, but IMPLICIT assumptions made here:
That space time is actually five dimensional.

No. The parameter s is the worldline parameter, proportional to the proper time s = c \tau. It is not a "real" extra dimension, in fact it is called "virtual" extra dimension. This is a mathematical trick.
It is very interesting because it shows a deep dualism of that theory with the Kaluza-Klein theory. If you suppose a compact worldline parameter with length \lambda_s, by discrete Fourier transform and by using \hbar you obtain a quantized mass spectrum M_n = \frac{n h}{\lambda_s c}; exactly as a compact time dimension with length T_t(\mathbf p) gives a quantized energy spectrum E_n(\mathbf p) = n h / T_t(\mathbf p). In fact, this relation evaluated in the rest frame (\mathbf p = 0) gives the Kaluza Klein tower: E_n(0)= M_n c^2, see figure.2.b. The Kaluza Klein tower is the energy spectrum evaluated at \mathbf p = 0. From this you can obtain the energy spectrum in a generic reference system by Lorentz transformation. Figure.2.a shows how the time periodicity transforms from the rest frame to a generic frame. This implies that every level of the energy spectrum follows the relativistic dispersion relation as shown in figure.2.b. That's why eqn.19 is the sum over a single index n (and not a sum over several n indexes, one for every compact dimension). Eqn.19 is also an infinite sum of Klein-Gordon lagrangians. All these energy eigenmodes are the harmonic modes of the same field with periodicity T_t(\mathbf p) in a generic frame and they can't be separated. They are part of the same fields and not different fields as in the Kaluza-Klein theory. This is the main difference between the two.

This would mean that the general Lorentz transformations belong to the group SO(1, 3 + 1) and this one has different irreducible representations from the ordinary SO(1, 3) group. Since the spin of the particles is exactly dependent on the representations of this group, the author could have at least given an effort to clarify this issue a bit more, since the whole subsequent derivation hinges on this fact;

The author has actually tried to clarify this point. Read the last paragraph of par.2.3.

( A five dimensional Dirac field is a sum of left handed and right handed four dimensional Dirac fields (with spin 1/2). This is because the fifth component of the five dimensional Clifford algebra \Gamma^M is \Gamma^5 = - i \gamma^5 ( and \Gamma^\mu = \gamma^\mu ). In that paper, however, only bosons are considered.)

That they only consider null geodesics dS^{2} = 0. I don't see any a priori reason for this and it is a huge constraint by itself. Saying the field is massless 5d is not an answer. The existence of a space-time metric between any two points should have nothing to do with the matter fields;

A four dimensional massless field lives on the four dimensional light-cone ds^{2} = 0, which is a lower-dimensional surface. Similarly a massless five dimensional field eqn.15 lives on the five dimensional lightcone <br /> dS^{2} = c^{2} dt^{2} - d\mathbf{x}^{2} - ds^{2} = 0<br /> which is a four dimensional surface. There, a virtual five dimensional massless fields is supposed, which is not really a five dimensional field. Since the compact extra dimension is the compact worldline parameter, you obtain the Minkowski metric ds^{2} = c^{2} dt^{2} - d\mathbf{x}^{2}.
Consider the first half page of par.1.2 a mathematical trick whose meaning will be clear later. The reason is to write an action that automatically take into account that the proper-time is compact for massive fields. This is the hardest part from a mathematical point of view, but it is definitively consistent even though it may seem artificial.

The term "compact dimension" could be somehow misleading. If so, just replace it with "periodic dimension".

That one of the dimensions is compactified. In doing this, you essentially break the Lorentz invariance of the theory.
I meant specifically the continuation of the last paragraph from p.8 on p.9 and the footnote 3. A quote:

(Are you reading the found. phys. version of the paper and do you mean footnote.4?)

It is well know in extra dimensional theories that compact dimensions don't necessarily mean that the Lorentz (or the gauge) invariance is broken. Consider eqn.4. This is a theory in compact space-time dimensions but it is Lorentz invariant as long as you consider that also the boundary must be transformed. Also Eqn.6 is four dimensional Lorentz invariant even if it is a sum over three dimensional actions. In fact, as long as all the modes n are considered eqn.6 is equivalent to eqn.4. The Lorentz invariance is broken only if you cut the tower, that is only if you consider a finite number of eigenmodes.

The basic characteristic of fields is that they can support excitations that propagate and carry certain momentum and energy.

Do you mean the momentum and the energy of the quanta of the field?

However, the number of excitations is essentially unlimited (except by the Pauli Exclusion Principle for fermionic fields). In this way, we might as well have two particle excitations moving in opposite directions. What is the rest frame of the field in this case?

If those particles are the quanta of the field, they must be off-shell to have opposite directions. Otherwise they are described by different field with different frequencies and wave-vectors. In a periodic field the unlimited particles are its harmonic excitations, that is its energy eigenmodes. But this eigenmodes are all on-shell. There is no violations of the energy conservation and it is not necessary to relax the classical laws of physics (that's why it is a deterministic theory). The rest frame is where all them have zero momentum, that is when the field (which is the sum all the nergy eigenmodes) has infinite spatial periodicity (infinite spatial compactification length).


I can try to provide further explanations if necessary.

 

[twofish-quant]

We must suppose that the coin rolls in the dark and suddenly a stroboscopic light is turned on for an infinitesimal instant.

But there has to be one and only one strobe light in the universe right? If you have people doing measurements with independent strobe lights, then the difference in timings between the independent strobe lights will introduce dynamics.

Since no hidden variables are involved, we are not in the hypothesis of the Bell theorem and the model can in principle satisfy the Bell inequality.

Yes. If there is a single strobe light in the universe, then Bell's inequality works. However, I think I can be clever and come up with another experiment that causes issues.

Also, doesn't this require simultaneity?

To resolve the deterministic mechanics of such a coin you need to measure time with a clock whose period is smaller than 10^-21 s (the emission of the polarized photon necessarily involved an electron). To have a precision of T*100000.1 you need a clock with periodicity faster than 10^-22 s in order to be sure to turn on the stroboscopic light at the right instant.

You don't. The period between pulses can be much larger than 10^-21s. The important part is that the pulse itself be timed to within 10^22 s. You can do this with some sort of mirror. If I take a mirror and have a laser bounce back and forth, the difference in A->B and B->A can be tiny even through the time it takes for the laser to travel A->B is large.

The difference between the cesium atomic clock and the de broglie clock of an electron is similar to the difference between a year and the age of the universe!

So build a clock based on two electrons, or one based on one proton and one electron. If you can use a de broglie clock of an electron to time the observation of another electron, then you should be able to think of some sort of experimental difference with QM.

Also it doesn't matter that the cesium atomic clock has a huge number of cycles, as long as the time between cycles is less than one fraction of the de broglie clock.

I fear that it is impossible to achieve this time resolution.

To convince people that the theory is correct, then you have to make it possible, and I don't see any physical reason why it won't work.

 

[twofish-quant]

Alice knows at T_1 what Bob will see at time T_2. It is clear now?

No. The problem is that Alice and Bob aren't talking to each other so neither knows at what precise time the other is looking at the coin.

Let me ask this question. Suppose Alice and Bob both look at the coin a difference in interval that is several times smaller that the deBroglie clock period. i.e. we can create an experiment so that A looks at the coin, and then B looks at it 1s +/- 1.5x10^-26 second later.

If Alice sees something, then would it be possible to determine exactly what Bob sees?

Also what is the decoherence time? i.e. how long does it take for A or B to take a measurement.

 

[twofish-quant]

One reason that I like the paper is that I have a soft spot in my heart for Kaluza-Klein models and so when I read the paper, that was the first thing I thought of.

One thing that I find nice about the paper is that if the entire universe has another time dimension which extremely small then you can communicate quickly between two different parts of the universe that seems spatial separated.

One crazy idea is that it's believed that cosmic inflation was causes when something happened to cause 3 space dimensions to suddenly get big. So it would be interesting to think about what the universe would look like if there were a lot of time dimensions that are tightly wound out, and then something happened to cause the universe to massive expand in the time direction.

 

[naturale]

No. The problem is that Alice and Bob aren't talking to each other so neither knows at what precise time the other is looking at the coin.

With rolling coin I mean flipping coin. Probably the misunderstood is originated by this.

I am absolutely not says that A and B talk with each other. The only thing they know could be their position w.r.t. the coin, which is the source of entangled photons.

To define even better this gedanken experiment we must suppose that
0) when the stroboscopic light it turned on, the coin will be forced to be orientated in such a way that it shows a face to Alice and the other face to Bob.

In this way what Alice and Bob see is entangled, they see two different faces of the same coin in S and T_0. In this case Alice knows what Bob will see (or have seen) without speak each other.

Let me ask this question. Suppose Alice and Bob both look at the coin a difference in interval that is several times smaller that the deBroglie clock period. i.e. we can create an experiment so that A looks at the coin, and then B looks at it 1s +/- 1.5x10^-26 second later.

If Alice sees something, then would it be possible to determine exactly what Bob sees?Also what is the decoherence time? i.e. how long does it take for A or B to take a measurement.

Now I try to replay to you specific question. First let me point out that 1s is an "eternity" with respect to the De broglie clock of an electron. The mass of the electron 9.10938215(45)×10−31 kg is known with a precision of 8 digits. To know the number of periods of a electron de Broglie clock in 1s we should know the mass of the electron with a precision of 20 digits, and we are quit far from that!

So we must imagine that:
1) if we know the mass of the electron with precision > 20 digits
2) if we have a clock with precision > 26 digits [the precision of the cesium atomic clock is 10 digits]
3) if the stroboscopic light doesn't introduce an additional element of randomness
4) if the coin in the first half period shows up (down) and in the second half down (up) to Alice (Bob)
5) if in a 1s you suppose to know exactly that the coin flips an even (odd) of half periods.
6) if during that "eternity" of 1s the coin is completely isolated, i.e. its periodicity is not changed by interactions or thermal noise
7) if ...
8) if ...
9) if ...
...

In this case I would say that if Alice sees up and coin flips an even (odd) of half periods, then Bob will see down (up). So to speak, the can see the classical evolution of the coin.

If some of the above conditions are not fulfilled (decoherence), the outcomes can be only described statistically through the usual laws of quantum mechanics. The generic Hilbert state defined before eqn.(31) in this case is |\phi &gt; = 1 / \sqrt{2} (|up&gt;\otimes |down&gt; - |down&gt; \otimes |up&gt;), where \otimes is the tensor product of two pure states of the periodic fields described in the paper.

But this is only my guess, it is how I interpret that theory.

The author demonstrates that the Feynman path integral eq.(40) "has been obtained just assuming relativistic periodic waves without any further assumption". To me this is sufficient to say that the theory reproduces ordinary quantum mechanics. If this is true, I believe it is true also because the paper has been peer reviewed, there is nothing more to say.EDIT:

Also what is the decoherence time? i.e. how long does it take for A or B to take a measurement.

From wikipedia :

"Quantum decoherence:

In quantum mechanics, quantum decoherence (also known as dephasing) is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior. Quantum decoherence gives the appearance of wave function collapse and justifies the framework and intuition of classical physics as an acceptable approximation: decoherence is the mechanism by which the classical limit emerges out of a quantum starting point and it determines the location of the quantum-classical boundary. Decoherence occurs when a system interacts with its environment in a thermodynamically irreversible way. This prevents different elements in the quantum superposition of the system+environment's wavefunction from interfering with each other."

I would say that in our case decoherence is a possible thermal noise such that one or more of the possible conditions above are not verified. Most likely condition 0) could be disturbed by the presence of thermal noise, other candidates are conditions 3), 4), 6), 7), 8), 9), ... .

 

[naturale]

One reason that I like the paper is that I have a soft spot in my heart for Kaluza-Klein models and so when I read the paper, that was the first thing I thought of.

One thing that I find nice about the paper is that if the entire universe has another time dimension which extremely small then you can communicate quickly between two different parts of the universe that seems spatial separated.

One crazy idea is that it's believed that cosmic inflation was causes when something happened to cause 3 space dimensions to suddenly get big. So it would be interesting to think about what the universe would look like if there were a lot of time dimensions that are tightly wound out, and then something happened to cause the universe to massive expand in the time direction.

That's true. Extra dimensions are between the most appealing theories of new physics on the market.

However, in that paper the fields are four dimensional fields, there are not "real" extra dimensions. That fields with intrinsic periodicity, which can be matched with ordinary second quantized fields, are "dual" to extra dimensional fields. The author says in the last paragraph that this is a "close parallelism with the AdS/CFT correspondence".

The theory is four dimensional, and I appreciate it because is physics that de Broglie, Einstein, Planck, Bohr & C could use at that time.
It is not simple to do good physics without playing with additional variables.

If you add an extra time dimension to reproduce quantum mechanics you end up to a theory with hidden variables and related no-go theorems.

 

[zonde]

Use a little bit of imagination. I meant that the stroboscopic light is tuned on and shows the coin at place S and time T_0 for an infinitesimal instant. Alice looks at a face of the coin and B looks at other side. Alice in A and at time T_1 (that is after a time delay T_1 - T_0 depending on how far she is from the coin) sees the face (the photon emitted by the that face of the coin at T_0 and S) with an up arrow (down arrow) whereas Bob in B sees the other face (the photon emitted by the other face at T_0 and S) with the down arrow (up arrow) at time T_2. Alice knows at T_1 what Bob will see at time T_2. It is clear now?

It seems clear what you mean but I don't see how this avoids Bell's inequalities.
From your description it follows that Alice sees (after period T_1 - T_0) classical variable that is produced at source - the face with an up arrow (down arrow) and the same for Bob (after period T_2 - T_0).

Please consider looking into this simple explanation of Bell's inequalities - https://www.physicsforums.com/showthread.php?p=2817138#post2817138"
ignoring of course the final conclusion as it is not conclusively proved by experiments.
This shows the problem with Bell's inequalities in very simple form. From this example it is clear that measurement of photon plays central role in this paradox and not exactly their production at source (this is just condition that we are free to adjust to our liking).

 

[zonde]

One thing that I find nice about the paper is that if the entire universe has another time dimension which extremely small then you can communicate quickly between two different parts of the universe that seems spatial separated.

What kind of topology is this where smaller distance in one dimension can dramatically reduce total distance even when distances in other dimensions are big?
Maybe I have poor imagination but it seems to me that this doesn't quite work. Well maybe if you introduce speed of some interaction as mathematical extra dimension.

 

[naturale]

It seems clear what you mean but I don't see how this avoids Bell's inequalities.

Do you see any hidden variable around in the above formulation of the EPR experiment?

From your description it follows that Alice sees (after period T_1 - T_0) classical variable that is produced at source - the face with an up arrow (down arrow) and the same for Bob (after period T_2 - T_0).


Please consider looking into this simple explanation of Bell's inequalities - https://www.physicsforums.com/showthread.php?p=2817138#post2817138"
ignoring of course the final conclusion as it is not conclusively proved by experiments.
This shows the problem with Bell's inequalities in very simple form. From this example it is clear that measurement of photon plays central role in this paradox and not exactly their production at source (this is just condition that we are free to adjust to our liking).

In deterministic physics, as in EPR-like experiments, the measurement doen't play such a central role. The Schrödinger's cat is dead or alive, independently if you watch it or not. Do you feel dead if I don't watch at you?

 

[DevilsAvocado]

Have a nice reading!

Thanks! I’m still "digesting", but I think the idea and conclusions are just beautiful!

Having not understood everything yet, there are some things that "bother" me. The first is that the word "wave function" is mentioned twice, in a footnote and in References. I’ll get back to that.

The next is related, and I see that you guys have already got into the subject, EPR-Bell experiments.

If you could explain EPR-Bell to me, I will probably send you a bottle champagne!

The crucial key here is that the entangled pair of photons shares the same wavefunction, and is outside each other’s light-cones when the wavefunction decohere. There is no way for communication. This is the way the author explains non-locality:

In the ’t Hooft approach to determinism as well as in the model with “stroboscopic quantization” there is the attempt to avoid local hidden variables. It is worth noting that the approach with compact time proposed throughout this paper has not local-hidden-variables that must be integrated out to get the quantum observables. We have just space and time coordinates which are physical variables. On the other hand the periodic conditions in eq.(3) can be regarded as an element of non locality (which is consistent with relativistic causality) in the theory. Therefore model proposed in this paper is deterministic²⁰ since it represents a possible way out of the Bell’s inequality or similar non-local-hidden-variable theorems [31].

If we take the very simple example (already mentioned) by Nick Herbert:

If both polarizers are set to 0º, we will get perfect agreement, i.e. 100% matches and 0% discordance.

To start, we set first polarizer at +30º, and the second polarizer at 0º:

If we calculate that discordance (i.e. the number of mismatching outcome), we get 25% according to QM and experiments.

Now, if we set first polarizer to 0º, and the second polarizer to -30º:

This discordance will also naturally be 25%.

Now let’s ask ourselves:

– What will the discordance be if we set the polarizers to +30º and -30º?

If we assume a local reality, that NOTHING we do to one polarizer can affect the outcome of the other polarizer, we can formulate this simple Bell Inequality:

N(+30°, -30°) ≤ N(+30°, 0°) + N(0°, -30°)​

The symbol N represents the number of discordance (or mismatches).

(The "is less than or equal to" sign is just to show that there could be compensating changes where a mismatch is converted to a match.)

We can make this simple Bell Inequality even simpler:

50% = 25% + 25%​

This is the obvious local realistic assumption.

But this wrong! According to QM and physical experiments we will now get 75% discordance:

sin^2 (60º) = 75%​

Now, I have 3 problems:

1) How can a deterministic model handle random and spatially separated apparatus?

2) How can the "periodic conditions" introduce non locality?

3) How can "relativistic causality" be compatible with what happen in EPR-Bell experiments? In one frame of reference Alice will cause the "decoherence", and in another frame of reference Bob will cause "decoherence"? (The shared wavefunction can only decohere once.)

 

[zonde]

Do you see any hidden variable around in the above formulation of the EPR experiment?

EPR paradox involves two non-commuting observables. Your formulation above involves only one observable so it has nothing to do with EPR paradox.
And yes I can see hidden variable in this formulation. Information is created at S when coin is illuminated and then this information is transferred to places A and B for observation. This information transfer is what is meant with hidden variables.

In deterministic physics, as in EPR-like experiments, the measurement doen't play such a central role.

Can't agree with this.

Did you look at Bell explanation I gave? Can we discuss it?

 

[twofish-quant]

In this way what Alice and Bob see is entangled, they see two different faces of the same coin in S and T_0. In this case Alice knows what Bob will see (or have seen) without speak each other.

That means however that there is one universal strobe light, which causes problems since you then have an absolute time coordinate.

First let me point out that 1s is an "eternity" with respect to the De broglie clock of an electron. The mass of the electron 9.10938215(45)×10−31 kg is known with a precision of 8 digits. To know the number of periods of a electron de Broglie clock in 1s we should know the mass of the electron with a precision of 20 digits, and we are quit far from that!

But you don't have to know the exact periods. You just need to know that they are the same.

In this case I would say that if Alice sees up and coin flips an even (odd) of half periods, then Bob will see down (up). So to speak, the can see the classical evolution of the coin.

OK, we are getting somewhere.

If some of the above conditions are not fulfilled (decoherence), the outcomes can be only described statistically through the usual laws of quantum mechanics.

Not really. In the pure situation we've agreed that you get predictions very different from QM. Now then the question is, how far away from the pure situation do you have to be before you see something different.

Also, personally, I think that they were hand waving when it came to Bell's inequality, and they didn't provide any particular information on how they intend to resolve it.

The author demonstrates that the Feynman path integral eq.(40) "has been obtained just assuming relativistic periodic waves without any further assumption". To me this is sufficient to say that the theory reproduces ordinary quantum mechanics.

If it's mathematically identical to quantum mechanics in all situations, then it's just another interpretation, and I'm not too interested in those, since those are a dime a dozen. The universal strobe light sounds a lot like the Bohm interpretation of QM.

If they have some alternative ideas that looks like QM at large scales, but is different in some situations, then it gets interesting.

If this is true, I believe it is true also because the paper has been peer reviewed, there is nothing more to say.

Peer reviewers are human and make mistakes. The threshold for passing peer review in these sorts of papers isn't very high (and that's a good thing IMHO).

I would say that in our case decoherence is a possible thermal noise such that one or more of the possible conditions above are not verified.

We know that the time it takes for something to go into a classical state is much shorter than the deBroglie period. We know this because if it takes longer to go into the classical state than the deBroglie period, then you won't be selecting a particular quantum state.

If it's thermal noise, then we know that there is some mechanism involving thermal noise that can cause things to happen on small fractions of the deBroglie period. Once we isolated whatever that is, we can change it to see if it affects quantum probabilities. For example, if it's thermal noise that cases things to decoher, then you can think of an experiment to see of temperature affects quantum probabilities.


Most likely condition 0) could be disturbed by the presence of thermal noise, other candidates are conditions 3), 4), 6), 7), 8), 9), ... .[/QUOTE]

 

[Dickfore]

If those particles are the quanta of the field, they must be off-shell to have opposite directions. Otherwise they are described by different field with different frequencies and wave-vectors.

Seems like you are simply using some 'vibe words' out of their proper context. I don't have the slightest idea what you are talking about.

 

[DevilsAvocado]

Shortly, imagine that the polarization of the two photons can be for instance determined (at the moment of the emission) by deterministic dynamics intrinsically too fast to be resolved experimentally (of the source)...

I have no idea if technically a polarizer can resolve dynamics faster than 10^-20 s.
From the physicsworld's article it seems hard to have such a resolution in time.

Then Alice and Bob see up and down or down and up, but if their resolution in time is poor they can predict the outcomes only statistically, and the probabilities are 50% and 50%. As Alice see the up arrow, she automatically knows that Bob will see the down arrow or vice versa. Since no hidden variables are involved, we are not in the hypothesis of the Bell theorem and the model can in principle satisfy the Bell inequality.

Use a little bit of imagination. I meant that the stroboscopic light is tuned on and shows the coin at place S and time T_0 for an infinitesimal instant. Alice looks at a face of the coin and B looks at other side. Alice in A and at time T_1 (that is after a time delay T_1 - T_0 depending on how far she is from the coin) sees the face (the photon emitted by the that face of the coin at T_0 and S) with an up arrow (down arrow) whereas Bob in B sees the other face (the photon emitted by the other face at T_0 and S) with the down arrow (up arrow) at time T_2. Alice knows at T_1 what Bob will see at time T_2. It is clear now?

With rolling coin I mean flipping coin. Probably the misunderstood is originated by this.

I am absolutely not says that A and B talk with each other. The only thing they know could be their position w.r.t. the coin, which is the source of entangled photons.

To define even better this gedanken experiment we must suppose that
0) when the stroboscopic light it turned on, the coin will be forced to be orientated in such a way that it shows a face to Alice and the other face to Bob.

In this way what Alice and Bob see is entangled, they see two different faces of the same coin in S and T_0. In this case Alice knows what Bob will see (or have seen) without speak each other.

...

In this case I would say that if Alice sees up and coin flips an even (odd) of half periods, then Bob will see down (up). So to speak, the can see the classical evolution of the coin.

In deterministic physics, as in EPR-like experiments, the measurement doen't play such a central role. The Schrödinger's cat is dead or alive, independently if you watch it or not. Do you feel dead if I don't watch at you?

naturale, with all due respect, I think you are missing one important factor here; EPR-Bell.

It seems like you are talking about the 1935 Einstein, Podolsky and Rosen 'vision' on the paradox, with only perfect correlation, and a presumed determinism.

But John Bell changed all that, and the measurement does play the ENTIRE role! And it is physically confirmed beyond any doubts that (deterministic) Local Hidden Variables (LHV) does not work. I know you are not talking about "Hidden Variables", but still you assume that the outcome is set at the source, and this is just impossible in practice.

In EPR-Bell experiments Alice do not always see the opposite side of Bob’s flip coin. When not aligned parallel for perfect correlation – they can get the same side, randomly!

The statistics of EPR-Bell experiments is sinusoidal. Deterministic models (at the best) always produce non-sinusoidal statistics:

And it all depends on the RANDOM settings of the polarizers, which are outside each other light-cone.

 

[naturale]

Thanks! I’m still "digesting", but I think the idea and conclusions are just beautiful!

Having not understood everything yet, there are some things that "bother" me. The first is that the word "wave function" is mentioned twice, in a footnote and in References. I’ll get back to that.

The next is related, and I see that you guys have already got into the subject, EPR-Bell experiments.

If you could explain EPR-Bell to me, I will probably send you a bottle champagne!

Yes, I was speaking about EPR experiment. The EPR_BEll experiment is pretty much complicated to figure out. As soon as I will find time I'll try to do what I can, at least from a formal point of view by using:

The generic Hilbert state defined before eqn.(31) in this case is |\phi &gt; = 1 / \sqrt{2} (|up&gt;\otimes |down&gt; - |down&gt; \otimes |up&gt;), where \otimes is the tensor product of two pure states of the periodic fields described in the paper.

Hopefully this should involve the "element of non-locality" of the theory as you suggested:

The crucial key here is that the entangled pair of photons shares the same wavefunction, and is outside each other’s light-cones when the wavefunction decohere. There is no way for communication. This is the way the author explains non-locality:

This non-lacality (periodic boundary conditions) actually is why the field can self-interferes. Moreover the probability of an observable eqn.41 is given by a field (the Born rule is mentioned in footnote n.16, is this your problem with the wave function?).
This yields to the same composition rule of probabilities of quantum mechanics P(a+b) \neq P(a) + P(b)..
If I'll ever find such an explanation I will write a paper first and then I'll ask you a battle of champagne (but I fear that to have that bottle of champagne you what something more than a formal description, something like an intuitive example, isn't it?).
At this point I have a request to you all: I would like that this tread is focused on the results discussed on that paper, which (IMHO) are extremely ex exciting.

From a mathematical level the paper seems to provide an exact matching with ordinary quantum mechanics by using essentially "deterministic" physics. I would like to discuss about this formal matching and its limits.

In few words I would like to discuss this:

Since we have inferred the Hilbert space eqs.(25), eq.(28) and eq.(29), the Schr ̈odinger equation eq.(27), eq.(30) and eq.(33), the commutation relations eq.(44), the path integral eq.(40), the Heisenberg uncertainty relation eq.(45) and the Bohr-Sommerfeld condition eq.(46) from the periodicity assumption, it is reasonably correct to interpret our theory as a quantum theory.

or

The effective time evolution of a periodic field is described by the Schr ̈odinger equation in a Hilbert space. Due to invariance by space-time translations of periods there are different classical trajectories with different winding number between the initial and final points. This gives rise to interference between different on-shell paths and thus to a path integral formulation, without relaxing the variational principle.

Other interesting aspects of the theory are described in the version 4 (arXiv:0903.3680v4). I believe the graphical interpretation of the path integral is something that should be read. The EPR paradox is not mentioned in that paper. The only think that is mentioned is that there are no hidden variables.

As long as someone of you do not prove that the Bell theorem can be adapted to rule out the case of intrinsic periodicity of the fields, the theory must be considered compatible with quantum mechanics (the Bell's hypothesis of local hidden variable can't be used in our case).