# Is quantum field theory really lorentz invariant?

1. ### Sam_Goldberg

46
Hi guys,

Before responding to my post, please note that I am only familiar with the mathematics of nonrelativistic quantum mechanics, and don't know any quantum field theory. All I have is this vague idea that quantum field theory is the union of special relativity and quantum mechanics, where the number of particles is not definite and both electrons and photons are described as excitations of a quantum field. The dynamics of the electrons and photons are, apparently, lorentz invariant.

Anyway, my question is this: there seem to be certain elements in nonrelativistic quantum mechanics that I can't see how to make lorentz invariant. No matter what interpretation one uses, it seems as if we have an irreparable violation of lorentz invariance. Take the Copenhagen interpretation, with its wavefunction collapse. The process of wavefunction collapse blatantly violates lorentz invariance, and I do not see how one could suitably modify it in such a way that it becomes lorentz invariant. Have they have figured out how to extended wavefunction collapse to quantum field theory such that it is lorentz invariant?

Even if one uses an interpreation without wavefunction collapse I don't see how one could still have lorentz invariance due to the nonlocality of quantum mechanics (EPR experiment). For example, in Bohmian mechanics, a many particle system is guided by the wavefunction. Since the wavefunction lives in configuration space, the guiding equation for the particles is nonlocal and in violation of lorentz invariance.

I am only familiar with the Copenhagen and Bohm interpretations, and I don't see how either can be extended to obtain lorentz invariance. Maybe quantum field theory does this, and I am simply not aware of it. Or it might be that there is an interpretation of quantum mechanics that is readily extendable to ensure lorentz invariance. To be honest, I am doubtful that a different interpretation of quantum mechanics will really make a difference. I have only heard of the many worlds interpretation and have not studied it in detail, but it seems that the branching of a universe into many other universes is also in violation of lorentz invariance (just like wavefunction collapse), and it does not seem easy to fix it. So I'm really stuck.

Or perhaps quantum field theory isn't lorentz invariant?

2. ### Demystifier

5,511
Despite nonlocality, there is a way to make Bohmian interpretation Lorentz invariant, even without quantum field theory. For a non-technical explanation of how is that possible, see
http://xxx.lanl.gov/abs/1002.3226 [to appear in Int. J. Quant. Inf.]
http://xxx.lanl.gov/abs/1007.4946

3. ### genneth

979
Measurement is inherently not covariant, because you need an observer, who then defines preferred time-slicing of spacetime. However, everything else is just fine --- the dynamics, etc. can be generated in such a way as to leave the measurement process as the only non-relativistic element in the theory.

4. ### dx

2,009
In quantum electrodynamics, the measurable quantites are field averages over spacetime regions, rather than fields at spacetime points. So the assumption of temporal ordering of measurements in non-relativistic quantum mechanics can no longer be upheld, since such an orering is possible only when the time intervals of the corresponding regions do not overlap. Thus, there are new features of complementarity of description involved.

Last edited: Sep 21, 2010

I would refrain from using the term "observer". It is philosophically misleading. Observer is not needed. Measuring and registering the results apparatus is needed. We can measure the temperature on the other side of the Moon even if no one is there.

6. ### Phrak

It seems that way to most. The Multi World Theory (MWT) proponents think they should get around this by replacing objective wavefuction collapse with subjective wave function collapse.

However, I am very sure that MWT is not required, and that the seeming nonlocality of quantum mechanics is false, as Einstein and Schrodinger would have it dispite J. S. Bell's ingenious argument.

Can you define Einstein locality?

Last edited: Sep 21, 2010

The process of wavefunction collapse blatantly violates lorentz invariance

Are you sure?

Classical interventions in quantum systems. II. Relativistic invariance
Authors: Asher Peres
(Submitted on 10 Jun 1999 (v1), last revised 7 Feb 2000 (this version, v2))

Abstract: If several interventions performed on a quantum system are localized in mutually space-like regions, they will be recorded as a sequence of quantum jumps'' in one Lorentz frame, and as a different sequence of jumps in another Lorentz frame. Conditions are specified that must be obeyed by the various operators involved in the calculations so that these two different sequences lead to the same observable results. These conditions are similar to the equal-time commutation relations in quantum field theory. They are sufficient to prevent superluminal signaling. (The derivation of these results does not require most of the contents of the preceding article. What is needed is briefly summarized here, so that the present article is essentially self-contained.) ​

8. ### Phrak

Do you see the deductive dissonance you've introduced in your argument? "Observer is not needed" is followed by a statement about "We" as observers--we who measure.

The distinction to be made is not between measuring devices and observes, but coherent and decoherent systems.

9. ### Demystifier

5,511

There is also a critique of this paper:
http://xxx.lanl.gov/abs/quant-ph/0109120 [Phys.Rev. A64 (2001) 066101]​

It is a very weak critique. The author is not even quoting all relevant papers, probably because he is not aware of them.

11. ### Demystifier

5,511
In a short comment on a specific paper, it is not appropriate to cite all indirectly relevant papers. Do you have a concrete objection on the arguments used in the Comment?

Yes, I do. He talks about "wave functions":

"In his approach (as well as in the approaches of many others), the wave function, described by quantum mechanics, is not a material object, but only a mathematical tool for calculating probabilities. and then uses density matrices."

Wave functions evaporated. No distinction is made between individual systems and statistical ensembles. The critique should be addressed differently.

Moreover, I wonder how the referee of this paper could let the main argument be published:

"Therefore, it would be a miracle if the unique nonrelativistic deﬁnitions of KA and LA would give the relativistic equation (7). It is not shown in [2] that this miracle happens. We have shown explicitly that this miracle certainly does not happen for N = 1."

Simply stated: "I can't show that this paper is wrong, but I strongly believe it must be wrong".

13. ### Fredrik

10,603
Staff Emeritus
The way I see it, "wavefunction collapse" is only needed if you make two unjustified assumptions on top of QM: a) that the wavefunction describes (i.e. represents the properties of) a physical system, and b) that there's only one world. If you don't make both of those assumptions (which are not part of QM by the way), you don't need to make a third assumption to try to solve the problems caused by the first two. (I also strongly doubt that the collapse axiom actually solves those problems, but that's another story).

The alternative to a) is to assume that the wavefunction represents the statistical properties of an ensemble of identically prepared systems (the ensemble interpretation/statistical interpretation) or to not make any assumptions at all (the "shut up and calculate" interpretation).

My point is that the collapse axiom has much bigger problems than Lorentz invariance, but I also think that it either makes both non-relativistic and special relativistic QM inconsistent, or neither. The "instantaneousness" of it all isn't a problem since it can't be used to send FTL messages.

14. ### Demystifier

5,511
No, simply stated it means the following: "I can't show that this paper is wrong, but the author (Peres) of the criticized paper also cannot show that his paper is right."
I think this is a sufficient reason for criticism of the Peres paper.

Also, there is a reply by Peres himself (not on the arXiv, but published in PRA), and his response to the criticism of his paper does not share your views at all.

So, if he can't prove that the paper is wrong, why to write a paper at all? As a joke? As a proof that the referees are asleep? Well, he succeeded.

As for Peres he does not have to share my views the same way I do not have to share his.

16. ### pellman

623
Wait. Measurable quantities are the only quantities which must be covariant. We don't necessarily require purely theoretical quantities to obey relativity or anything else (e.g., conservation laws) as long as the predicted observable quantities do since theoretical quantities may only be artificial and not correspond to anything in reality at all.

You always have to choose a particular reference frame to compare with experiment/observation, whether classical or quantum. But your measured quantity better transform to other frames as some representation of the Lorentz group.

17. ### Demystifier

5,511
Measurable quantities are covariant, but what may NOT be covariant is a PHYSICAL PROCESS (like wave-function collapse) that gives measurable quantities definite values.

Of course, if there is no such process (no-reality interpretations of QM), then there is nothing to be non-covariant.

18. ### Demystifier

5,511
To point out some weaknesses of the Peres paper, to motivate him (Peres) to better develop his theory in a next paper?

That is a good subject for a private email but not for a paper in a peer-reviewed journal with a high impact status. Nicolic did not do his homework and reports his failure. The referees did not do their job.

20. ### Sam_Goldberg

46
Hi Demystifier,

I read your suggested article on Bohmian relativistic QM and I'm still not convinced. Nicolic defines a parameter s for the world lines of all the particles in the system, but the question of how to make the parametrization seems ambiguous. It is clear that different parametrizations produce different results. Even if we say ds = d(proper time) for each particle, it is still unclear at which point in a given particle's world line we say s = 0. And which point we select matters.

Next, for everybody, I have a question on the Copenhagen interpretation's dealing of the EPR experiment. I would like to recall Einstein's actual quote: "If, without in any way disturbing the system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."

If I understand this, it would mean that when we make a measurement and go from a superposition of spin-correlated states |u,D> + |d,U> to a single one, that is, either |u,D> or |d,U>, then even though our measuring device is near the lowercase particle, since we know the value of the upercase particle's spin with certainty, it's a variable with physical reality. But this would mean that the physically real quantities are not lorentz invariant!

One could refute this argument by saying that we did not directly measure the uppercase particle's spin and that therefore it never gained any physical reality. This would be saying that Einstein's criterion of reality is wrong. Then the physically real quantities are lorentz invariant. So the lesson is that only what we directly measure gains physical reality, and further, that the physically real quantities must be covariant.

I feel, however, that there is a problem with the idea that only what we "directly" measure gains physical reality. In the Copenhagen interpretation, we make a cut between our quantum system and our classical measuring apparatus, and where we place the cut is, according to Bohr, arbitrary. Let's say we are measuring the position of an electron. If we say that only the electron is part of the quantum system, then after the measurement, the electron position is a physical reality, since we directly measured it. However, let's say we include a pointer (that correlates with the electron position) in the quantum system as well. Then in the new system-apparatus framework, the electron and the pointer are entangled, and we "directly" measure not the electron, but the pointer. It is clear that the pointer gains a position, and this is a physical quantity. However, this situation is very similar to EPR, and with similar reasoning, one would have to say that the electron did not gain a position. One would have to say that only when we "directly" measure the electron it gains an element of physical reality.

So this is really problematic. If we assume that the particle we don't directly measure gains an element of physical reality (as Einstein says), then we must say that physical quantities are not lorentz invariant. But if we assume that the particle we don't directly measure does not gain an element of physical reality, then we have the problem mentioned in the previous paragraph: by placing the cut between system and apparatus differently, our electrons will never gain definite positions, spins, etc.