# No logically consistent relativistic quantum theory?

1. Aug 2, 2012

### genericusrnme

I just read this as I was skimming over the preface of a relativistic QM book. My question is this; is it the case that there is not yet a logically consistent and complete relativistic quantum theory or is this statement just because the book was published in 1980 and there has since been progress made?

2. Aug 3, 2012

### DrChinese

When talking about quantum theory, the terms "complete" and "logically consistent" end up being problematic and essentially in the eyes of the beholder. I don't believe it is possible to answer your question as asked. There are relativistic treatments of QM preceding 1980, and there has also been substantial progress since.

3. Aug 3, 2012

### ZapperZ

Staff Emeritus
One of the things that we try to instill in this forum is that people make full citation of their sources. We want to turn this into a habit.

https://www.physicsforums.com/blog.php?b=2703 [Broken]

What you have written here is highly insufficient to carry on a proper scientific discussion. So please cite completely your source, i.e. Author/s of the book, title, date of publication, etc.

Zz.

Last edited by a moderator: May 6, 2017
4. Aug 3, 2012

### vanhees71

The statement for sure is wrong, since there is a very consistent relativistic quantum theory, called local relativistic quantum field theory, of which the most prominent example is the standard model of elementary particle physics. It's successfulness is even enerving since we hope to find evidence for "physics beyond the standard model" to solve some physical problems like to figure out the nature of "dark matter" in the universe or the hierarchy/finetuning problem from the Higgs-mass renormalization. Since July 4th we can however be pretty sure that the standard model is confirmed in all its aspects: All particles predicted by this model are observed including all their properties (except for the Higgs itself, where after its very probable discovery at the LHC the real work only begins: namely to figure out its properties and whether they are consistent with the predictions of the standard model or (hopefull?) not).

On the other hand, we have no satisfactory quantum description of gravity or spacetime itself, and in this regard our understanding of nature in terms of (relativistic) quantum theory is incomplete. Spacetime/gravity is still described in a classical manner, while everything else seems to be very well described by quantum field theory. In this respect we have not a comprehensive relativistic quantum theory, but the one we have is logically consistent.

5. Aug 3, 2012

### genericusrnme

I just read back over the preface this morning with fresh eyes and I saw the phrase 'The lack of complete consistency in this theory (QED) is shown by the occurance of divergent expressions when the mathematical formalism is applied directly'. I think I skipped that whole part since it went over onto a new page. I have heard about these divergencies before although I don't know anywhere near enough about the theory to know where they come from..
So I guess I just answered my own question!

Landau and Lifgarbagez - Relativistic Quantum Mechanics, second edition, published ni 1982

I already knew about those problems but I thought they were a little 'higher up', the way I read it made me think they saw some deep structural flaw. Although from my first reply it seems I read it wrongly.

Last edited by a moderator: May 6, 2017
6. Aug 4, 2012

### vanhees71

The infinities in QED and all other quantum field theories are well understood and resolved by modern renormalization theory. Thus I think they are not an argument against QFT in the sense that it might be inconsistent.

7. Aug 4, 2012

### strangerep

Leaving aside the lack of existence proofs for most of the interesting 4D QFTs, there remain puzzles like quantum triviality. I.e., for some nontrivial classical theories, canonical quantization and orthodox renormalization leads to a trivial quantum theory. J.R. Klauder has been doing some interesting work on this in recent years: he advocates using different (classical) commutation relations as a starting point -- he calls it "affine quantization" -- and a space of correspondingly different coherent states. He gives examples where canonical quantization leads to triviality but affine quantization does not. A recent relatively-readable review is this paper:

J. R. Klauder,
"Enhanced Quantum Procedures that Resolve Difficult Problems",
Available as arXiv:1206.4017

Abstract:
A careful study of the classical/quantum connection with the aid of coherent states offers new insights into various technical problems. This analysis includes both canonical as well as closely related affine quantization procedures. The new tools are applied to several examples including: (1) A quantum formulation that is invariant under arbitrary classical canonical transformations of coordinates; (2) A toy model that for all positive energy solutions has singularities which are removed at the classical level when the correct quantum corrections are applied; (3) A fairly simple model field theory with nontrivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (4) A model of scalar field theories with nontrivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (5) A viable formulation of the kinematics of quantum gravity that respects the strict positivity of the spatial metric in both its classical and quantum versions; and (6) A proposal for a nontrivial quantization of $\phi^4_4$ that is ripe for study by Monte Carlo computational methods. All of these examples use fairly general arguments that can be understood by a broad audience.

8. Aug 5, 2012

### Fredrik

Staff Emeritus
"Logically consistent" is a strange choice of words. Either there is a theory, or there isn't one. It's not like we have a bunch of theories that are all logically inconsistent. (A logically inconsistent "theory" wouldn't make unambiguous predictions). I will also ignore the word "complete" because I don't know what it means here. So I will answer the question "Is there a relativistic quantum theory?" first, and then I will try to guess what you really wanted to ask.

The simple answer is yes. There are lots of them. They just aren't very interesting. The simplest ones describe a universe that's completely empty except for a single particle of a given type. Such a theory can be defined using quantum fields, so we can also say that there are many relativistic quantum field theories.

So the question is actually kind of trivial. What we really should be asking is if there are any relativistic quantum field theories with interactions.

The answer depends to some extent on what exactly we mean by "theory" and "relativistic quantum field theory with interactions". I would say that a theory is defined by a set of assumptions that can be used to make predictions about results of experiments. QED is strictly speaking not a theory according to this definition, because of problems with divergences and stuff. But physicists would certainly not agree that they don't have a theory of electrons, positrons and photons. A simple way out of this problem is to define, for each positive integer n, QED(n) as the idea that we should use the usual QED techniques (in particular Feynman diagrams) but discard all Feynman diagrams of order higher than n. Since each choice of n leads gives us unambiguous predictions about results of experiments, each QED(n) is a theory by my definitions.

Now the question is, should a theory like QED(2) be considered a "relativistic quantum field theory with interactions". I think most physicists would say no. It's an excellent theory of physics, but it's not a relativistic quantum field theory. It's also mathematically ugly, since we're throwing away terms for no other reason than that this gives us a well-defined theory of physics.

So what is a quantum field theory with interactions? I don't think there's a definition that everyone is satisfied with, but I believe the most widely accepted definition is the one that says that the Wightman axioms must be satisfied. I think the problem with QED is that no one knows a way to define the fields that ensures that the Wightman axioms are satisfied. (But my knowledge of this is rather poor, so I could be wrong about the details). There's a million-dollar prize waiting to be collected by the first person to find a theory that satisfies the Wightman axioms, and involves fields that are similar to those in QED. The Wikipedia page I linked to has more information about that.

The prize is specifically for Yang-Mills theories, like QED. One can also ask if there are any other relativistic QFTs with interactions. Again, my knowledge of the subject is poor. I don't know if the answer is yes or no.

Last edited: Aug 5, 2012
9. Aug 5, 2012

### atyy

I think there are rigourous relativistic QFTs in 2D and 3D. I think it is is unknown for 4D, but conjectured to be true for Yang-Mills theory.

http://www.arthurjaffe.com/Assets/pdf/CQFT.pdf:

"With these methods, one could establish the first non-trivial example of the Wightman axioms. Viewed differently, this body of work established the mathematical compatibility of quantum field theory with special relativity. While this work only applied in two-dimensional space-time, it marked the crossing of a major set of obstacles barring progress."

"led to the first example of a non-trivial Wightman theory on M3."

"Can one find a non-trivial, non-linear quantum field in four-dimensional space-time? The most promising candidate for a non-trivial and physically-interesting field theory on Minkowski 4-space is the Yang-Mills theory with an SU(2) gauge group."

There has also been major progress as to why QFTs don't have to exist at all energies in order to produce meaningful predictions at low energies. This is the effective field theory framework. http://arxiv.org/abs/hep-th/0701053

The standard model of particle physics is believed to be only an effective field theory at best (unless it turns out to be "asymptotically safe", which has neither been proved nor disproved).

Last edited: Aug 5, 2012