# Is quantum field theory really lorentz invariant?

by Sam_Goldberg
Tags: field, invariant, lorentz, quantum, theory
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 P: 1,411 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.
 P: 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.
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P: 4,591
 Quote by Sam_Goldberg 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.
There is no ambiguity, in the sense that the trajectory in spacetime does not depend on s at all. See e.g. Eq. (30) in
http://xxx.lanl.gov/abs/quant-ph/0512065

 Quote by Sam_Goldberg Even if we say ds = d(proper time) for each particle,
We do not have a freedom to say that. It can be shown that s is NOT the proper time, but rather a generalized proper time. See the appendix in
http://xxx.lanl.gov/abs/1006.1986

 Quote by Sam_Goldberg it is still unclear at which point in a given particle's world line we say s = 0. And which point we select matters.
That is true, but this property is a virtue, not a drawback. When we are ignorant about the exact spacetime position at s=0, all we can say is that the probability of a given spacetime point at s=0 is given by |psi|^2. In this way deterministic Bohmian mechanics reproduces probabilities from the "standard" purely probabilistic interpretation.
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P: 4,591
 Quote by arkajad That is a good subject for a private email but not for a paper in a peer-reviewed journal with a high impact status.
Well, if you insist, there is always one additional reason to publish a paper in a peer-reviewed journal with a high impact status: It is the PUBLISH OR PERISH principle.
And this principle is not invented by those who obey it.
 P: 1,411 I know the principle. I used to be guilty of this myself. It is partly because of this principle that we are really reading less and less papers. Comparing the quality of so called scientific publications 100 years ago and now - the emerging picture is not very encouraging. But that's another subject. Good thing is that now I have Nicolic's "Superluminal velocities" paper on my desk and it looks like it's gonna be an interesting read.
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P: 4,591
 Quote by arkajad I know the principle. I used to be guilty of this myself. It is partly because of this principle that we are really reading less and less papers. Comparing the quality of so called scientific publications 100 years ago and now - the emerging picture is not very encouraging. But that's another subject. Good thing is that now I have Nicolic's "Superluminal velocities" paper on my desk and it looks like it's gonna be an interesting read.
With that I agree!
P: 46
 Quote by Demystifier When we are ignorant about the exact spacetime position at s=0, all we can say is that the probability of a given spacetime point at s=0 is given by |psi|^2.
Is this really true? I have read in the Bohm and Hiley book why the probability density approaches |psi|^2 in nonrelativistic quantum mechanics, and it is due to the chaotic dynamics in most physical situations. I would really like to see a proof the statement you just made for relativistic quantum mechanics, or at least some intuitive hand-waving that will help me understand.

Furthermore, if the probability of a given spacetime point at s=0 is |psi|^2, then what is the probability of a given spacetime point at s=1 or s=2? I seriously doubt that it is also given by |psi|^2. But then we have a problem, since the time evolution of psi is given by the Klein-Gordon or Dirac equation, which don't have s anywhere. Then, wouldn't the probability for every value of s be |psi|^2?
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P: 4,591
 Quote by Sam_Goldberg I have read in the Bohm and Hiley book why the probability density approaches |psi|^2 in nonrelativistic quantum mechanics, and it is due to the chaotic dynamics in most physical situations.
This is one possible explanation, but not the only one.

 Quote by Sam_Goldberg Furthermore, if the probability of a given spacetime point at s=0 is |psi|^2, then what is the probability of a given spacetime point at s=1 or s=2? I seriously doubt that it is also given by |psi|^2.
But it is.

 Quote by Sam_Goldberg Then, wouldn't the probability for every value of s be |psi|^2?
Exactly!
P: 196
 Quote by Sam_Goldberg 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 ?
The question to what it comes down to: can measurements between two space-like seperated points causally affect each other? In QM that is very well possible, in QFT it is not, which makes the theory lorentz invariant. Read chapter two in the standard QFT text of Peskin and Schroeder, where it is very well explained (you can skip the math if you like, they explain it also nice in words).

That the collapse is instaneously if you mak one measurement (which is the case in both QM and QFT!), is not relevant for the theory to be relativistic or not. But is faster than light communication between two events/ measurement possible or not, that matters!
P: 1,411
 Quote by kexue which makes the theory [QFT] lorentz invariant
and, one should add, as Feynman has noticed, for non-trivial interactions, mathematically inconsistent.
 P: 46 Professor Nikolic, I have just one more question. In your papers you emphasize that the relativistic wavefunction is a function on the 4n dimensional configuration space. However, in nonrelativistic mechanics, it appears as if the wavefunction lives in 3n dimensional configuration space and evolves in time according to the Schrodinger equation. Specifically, at low energies it appears as if the wavefunction on 4n dimensional configuration space has a nonzero value only when the time coordinates for all the particles are equal, and I'm still not clear how that mathematically arises. I still don't have a clear picture how the relativistic case reduces at low energies to nonrelativistic Bohmian mechanics. Thanks for helping me out.
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P: 4,591
 Quote by Sam_Goldberg Professor Nikolic, I have just one more question. In your papers you emphasize that the relativistic wavefunction is a function on the 4n dimensional configuration space. However, in nonrelativistic mechanics, it appears as if the wavefunction lives in 3n dimensional configuration space and evolves in time according to the Schrodinger equation. Specifically, at low energies it appears as if the wavefunction on 4n dimensional configuration space has a nonzero value only when the time coordinates for all the particles are equal, and I'm still not clear how that mathematically arises. I still don't have a clear picture how the relativistic case reduces at low energies to nonrelativistic Bohmian mechanics. Thanks for helping me out.
That is a good question. The answer is that even nonrelativistic QM can be formulated in the 4n-dimensional configuration space. I have explained that in some of my papers, but this result is in fact much older than I am. For an old reference see
S. Tomonaga, Prog. Theor. Phys. 1, 27 (1946).
The usual single-time nonrelativistic QM is just a special (coincident) case of the more general many-time nonrelativistic QM.
For example, if you want to calculate the expectation values of products of observables measured at DIFFERENT times, the many-time formulation of nonrelativistic QM is a very convenient way to do this.

By the way, thanks for not writing my name as "Nicolic", which for some reason many people do. :-)
P: 1,411
 Quote by Demystifier By the way, thanks for not writing my name as "Nicolic", which for some reason many people do. :-)
Sorry for that. I know the pain.
 Sci Advisor P: 4,591 By the way, if you write "Nicolic" in the google, the first thing it writes is: "Did you mean: Nikolic"
P: 582
 Quote by Sam_Goldberg Specifically, at low energies it appears as if the wavefunction on 4n dimensional configuration space has a nonzero value only when the time coordinates for all the particles are equal, and I'm still not clear how that mathematically arises.
Well, whatever it is, it must be the case that in the limit for c goes to infinity (or small energies) the wave function for a single particle system satisfies

$$\frac{\partial}{\partial t}\int{|\psi(x,t)|^2 d^3x}\rightarrow 0$$

so that we can normalize the integral over space and use the wave-function-squared as a probability density at each time t, so that time reduces to a parameter.

For multiple particles, you would have to express the wave-function in a many-time formulation, as Demystifier says, so that the wave-function would represent the probability amplitude of observing the first particle at x1,t1, the second particle at x2,t2, etc. The non-relativistic limit would be letting c go to infinity and setting t1 = t2 = ... = tn = t. But the wave-function is not necessarily zero if the times are unequal. It's just that you can only restore the usual time-parametrized Schrodinger picture by setting them equal.

I'm glossing over some fine points, but that is essence of it .

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