Philadelphia, PA

Dear Wechsler, 

In general terms, I would assume that it would be o.k. to make use of non-relativistic QM where attention to relativistic effects is not needed or not of interest or relevance to the system under study, that is, wherever it may reasonably be expected that relativistic contrasts of frames of reference do not come into play.

In general terms, special relativity can be thought of as a theory of measurement, or as implying a theory of measurement--according to which there is no absolutely privileged frame of reference. But it is known, for instance, that some very short lived particles, coming down through the atmosphere can only be detected at all because of their very high velocities --a contrast between their frame of reference and that of the observer on the ground. 

QFT covers the more general case,--or when in doubt? Being more specific may require detailed accounts of expected parameters. 

H.G. Callaway

If the kinetic energy is much less than the rest energy, then the non-relativistic approximation is appropriate; when the kinetic energy becomes comparable to the rest energy, that's when relativistic effects can't be neglected. 

Dear users,

Please refer also to my sub-question about reference frame. For a particle possessing rest-mass, the most natural frame of reference is the one in which it is at rest. But in such a frame its wavelength is infinite. So, for the QM is not a good frame to work with.

Moreover, if we deal with a system of particles, we have to examine all of them in the same frame. How do we choose this frame? Should we choose the center-of-mass frame? Should we choose the frame in which the source is at rest? Should we choose the frame in which the detectors are at rest?

There's a misunderstanding here: at the rest frame of a massive particle, its momentum is zero-but this is a statement about the average value of the momentum, not its higher moments. So it's a perfectly good frame. For a free particle, on an unbounded domain,  it simply means that one must use wavepackets-and that's done in all textbooks on quantum mechanics.

In any case what frame is used doesn't matter, since it's possible to pass from one to any other. So for a multiparticle state it doesn't matter, either-for certain processes the center-of-mass frame is convenient, for others the ``lab'' frame, where the detectors are at rest, is  more convenient. No physical quantity depends on the choice of frame, and quantities that do depend, only make sense as tools for computing quantities that don't. 

Stam, my friend,

As I recall what you and Partha Ghose told me, we have a problem with passing from a frame of coordinates to another one. This is why I ask how to choose the frame of coordinates in which to do our treatment.

You both told me that for passing from one frame to another we need the formalism of QFT, and this formalism doesn't accept a fixed number of particles. So, if we work with an entanglement of two particles, we have a problem.

Do I remember correctly what you said?

On the other hand, if the relative velocity of the frames is low, can we just apply to the wave-function the Lorentz transformation, and still pass from one frame to another without using QFT, and without the indetermination in number of particles? Could it be that at low velocities, these complications are negligible?

Please see, Dirac's wave-function for fermions passes from one frame to another without using QFT. So, why should we need QFT in passing between frames with bosons? It's an issue that bothers me a lot and I don't feel sure on it.

With thanks in adavance,

Sofia 

The most obvious answer is: you have to use QFT if you consider processes in which particle number is not constant. So, under the conditions you describe, you don't need QFT. You use the reference frame in which the equations describing the process under consideration take the simplest form. To give an example: if you consider particles enclosed by a container, then you choose the frame in which the container is at rest. Of course, there is a non-relativistic version of the Lorentz-transformations. That Fock-space methods for multi-particle systems work both in QFT and in non-relativistic many particle physics is best explained in the classical textbook S.S. Schweber: An introduction to relativistic quantum field theory.

Dear Ulrich,

Thanks for your clear answer.

Now, the non-relativistic Lorentz transformation is the Galilean transformation. Though, I ask about proper Lorentz transformations, which impose different time axes to frames in relative movement, i.e. the concept of simultaneity is affected. Though, the differences are not big, i.e. we don't have highly relativistic particles.

Can we use in this case relativistic QM, though not QFT?

With thanks,

Sofia

Dear Sofia

Of course you can also use relativistic quantum mechanics which employs 'real' Lorentz-transformations instead of their Galilean similes. However, relativistic quantum theories of systems with a fixed  number of constituents exist in various flavours and are physically not really satisfactory (I wrote my habilitation work on such theories). So, unless you have very good reasons to consider the kinematical differences between Lorentz-transformations and Galilei-transformations as vital in your context, you better stay within the unambiguous non-relativistic form of quantum mechanics.

It depends on the mass of the particle under consideration. Compton wavelength (h/mc) is a function of the rest mass (say m) of a particle. Below this length scale particle-antiparticle pair creation becomes an appreciable effect. One has to treat the dynamics (of an unspecified number of particles) in the framework of quantum field theory.

Oh, Ulrich !

Please tell me more!

Different people told me that relativistic QM with a fixed number of particles is problematic, but until now I don't have a clear physical picture, why so. WHAT happens when passing from one relativistic frame to another one? Why it is not good to do this transformation when we have a fixed number of particles, even at not so high energies?

A nice user indicated me material, but it was impossible to read it. And I remained with the idea that we have a problem, without understanding why, i.e. what happens. I would be very glad if you'd explain.

With thanks,

Sofia

Hi Sofia,

unfortunately I'm presently under time pressure so for a first information I would suggest that you look in my RG profile, contributions, the item: 'How to access all my journal articles'. There are two are articles, one with 'no-go theorem' in the title and one with 'relativistic  n-particle' in the title. These are my contributions to the topic. In short, the problem with relativistic quantum mechanics is that it is difficult (i.e enforces a complicated structure of the Hamiltonian) to satisfy the cluster decomposition property which is nearly trivially satisfied in a non-relativistivc multi-particle theory with pair interaction. Of course there is more recent material by others, I don't observe the scene any longer. It's a dead end as I see things today.

Dear Ulrich,

Yes, I noticed those articles of yours and I intend to read them. One of them, I asked for the full text, but I understand from you now that there is an easier way to get into your articles. I will try it. 

The issue is of top relevance for entanglements, which typically work with a fixed number of particles. There are works of great importance which examine the entanglements according to frames in relative movement.

Many thanks for your kindness.

Sofia

Dear Sofia,

on RG there is Eugene V. Stefanovich who is 'fully in the business of relativistic quantum dynamics'. On his profile page he has a lot of relevant texts for download. The one I know a bit (and which I admire) is Full-text.o504062v17. In Ch. 6 there is the cluster separation problem explained. You would probably benefit from contacting him.

There is a text by others with related intent:

B.D.Keister and W.N.Polyzou: Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics which floats somewhere in the web for free download. Here the three-body problem is treated in some detail.

For your two-body problem you fortunately may ignore all the subtle aspects of relativistic quantum theory.

Oh, Ulrich !

A lot of thanks! It's for quite a time that I have the feeling that I walk on unsure ground with my entanglements Yes, I'll follow your advices. 

With kind regards,

Sofia

Sofia, you are welcome.

Dear Ulrich,

I already read Stefanovich's chapter 6 from the book you recommended. For 2 particles that fly far away from one another we don't have the clustering problem. But, I am trying to contact him as you said that he is a specialist in relativistic QM.

With thanks and kindest regards,

Sofia

Both theories, relativity and Quantum, can be covered by using Distribution theory (see my work here in researchGate). impediments can be avoided by using the theory of distributions, while generalised functions are valide for infinite values too and an asymmetrical time interval of events (such as mass genesis) is permitted.

Furthermore by using distribution theory all physical laws, if relativity or quantum laws, can be handled as one single theory. Even if the laws fail in singularities[1], it is appropriate to use the adequate distribution theory, which works on discrete spaces[2] and on singularities too. Test functions have very good properties: infinite derivable, defined on compact spaces; transport of derivative from one factor to an other is allowed

[1] beginning singularities (like big-bang or steady state where from matter evolves) ending singularities (like black holes where matter finalises).

[2] in a fractal coordinate system there exist derivatives/ integration of real degree/order (not an integer). for discrete spaces  there are used set functions and the distribution theory.

Kinetic energy of electron in classic QM is equal to the energy at rest Eo=0.51MeV at the speed v=1.47c and it isn' t compatible with SR. In FQT instead the relativistic kinetic energy is equal to the energy at rest at the speed v=0.87c and it is compatible with SR. Now nevertheless this limit value of speed v=0.87c has to be interpreted correctly, in fact the non-relativistic approximation is valid when the kinetic energy is much greater than the rest energy, and not when the kinetic energy is much less than the rest energy, like it was written in a preceding comment, Consequently the non-relativistic theory is valid for high speeds (for v>>0.87c) and it isn't valid for low speeds (v0.87c).

Dear Sofia D. Wechsler and Ulrich Mutze

I was surprised to find out that you were discussing my book here 2 years ago. Ulrich, thank you very much for your kind words about my contribution. Long ago I learned relativistic quantum theory from your brilliant papers. So, I am very much in debt.

I agree with you that the multi-particle relativistic quantum mechanics of the Bakamjian-Thomas type is a dead end. The weak point of such theories was that they tried to satisfy relativistic invariance by building interaction operators as functions of relative positions and momenta. These operators are quite complicated, and in many-particle cases ensuring the cluster separability becomes almost impossible. I think than Wayne Polyzou and Bill Klink have not lost hope, but it my opinion there is no simple way to generalize this approach to systems with variable number of particles.

But I also believe that there is a bright future for a particle-based theory. This is the "dressed particle" approach, which I promoted in the book. It takes advantage of few nice features of quantum field theory. First, QFT is Poincare-invariant (see Weinberg). Second, the condition of cluster separability is very simple in QFT (see also Weinberg): the theory is separable if interaction terms (polynomials in creation and annihilation operators) have smooth coefficient functions in the momentum space. Of course, QFT has one repulsive property - the divergences, but the idea of the dressed particle approach is to reformulate QFT, so that its good properties (Poincare invariance, cluster separability, exceptionally accurate predictions) are preserved, but bad properties (self-interaction of particles, divergences) are eliminated. This can be done by means of the unitary dressing transformation first invented by Greenberg and Schweber in 1958.

As a result, we get a theory of particles with direct interactions. This theory looks similar to the Bakamjian-Thomas approach, but doesn't share its defects. Interactions are polynomials in creation and annihilation operators depending on particle momenta (so, complicated relative variables do not show up). The theory is Poincare invariant and cluster separable (by construction). There is no problem in describing processes changing the number of particles. The Hamiltonian is finite (no usual counterterms, cutoffs, etc), and the S-matrix coincides (by construction) with the usual S-matrix of renormalized QFT. So, the electron's magnetic moment, the Lamb shift etc. have the same accurate values as in QFT.

Sofia, I would be happy to answer your questions if you still have any.

Eugene.