Literature for QM to QFT step-by-step

[Gedankenspiel]

Hi all,

can anybody help me to find literature that takes the reader on a step-by-step path from non-relativistic quantum theory to relativistic quantum theory?

I imagine something like that: it starts with a single harmonic oscillator, analyzes the non-harmonic oscillator (does the state-space change somehow?), then goes on with coupled harmonic oscillators or free quantum fields, arriving at coupled non-harmonic oscillators, both non-relativistic and relativistic. A similar path could be followed from non-relativistic quantum mechanics via non-relativistic multi-particle physics to relativistic multi-particle physics. Just to give a rough idea.

The reason is: I am just wondering where in the line from one-particle QM to relativistic QFT the difficulties enter. Is it the multi-particle aspect? Or the interaction aspect? Or the relativity aspect? All combinations of these aspects could be carefully analyzed.

Any suggestions of literature are appreciated!

 

[Avodyne]

Any QFT text will do this in one way or another. My personal favorite is Srednicki.

 

[Gedankenspiel]

Thank you Avodyne for you fast reply.

I know Srednicki but he does not pinpoint exactly where the problems arise. For my purpose, he covers too much material but not nearly enough details of the foundations.

I am looking for a detailed treatment of the foundations of QFT starting at non-relativistic QM.

 

[George Jones]

What do mean by "difficulties" and "problems"? How can non-relativistic theory be the starting point for a relativistic theory, e.g. Newtonian gravity is not the starting point for general relativity.

My recommendation:

"Quantum Theory for the Gifted Amateur" by Lancaster and Blundell,

https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

Here, "gifted" means "has the equivalent of an undergraduate physics education" and "amateur" means "does not necessarily plan on becoming a professional quantum field theorist". The first few chapters (each chapter in the book is short) are about non-relativistic stuff. Part III "The Need for quantum fields" occupies chapters 8 - 15. The non-relativistic Schrodinger equation is produced by an appropriate limit in part III.

 

[Gedankenspiel]

Thanks George! I'll have a look at this book.

General relativity does not start at Newtonian gravity because it introduces a completely new concept. In contrast, relativistic quantum theory is based on the same principles of quantum theory as non-relativistic quantum theory is. So I wonder if it is possible to gradually extend well understood theories to finally end up with a relativistic quantum theory. And if not, where exactly the problems arise.

With "difficulty" I mean the fact that, as far as I know, no rigorously defined relativistic quantum theory in 3+1 dimensions is known. But exactly why?

 

[Avodyne]

Short answer: because the cutoff cannot be taken to infinity in a theory that is not asymptotically free. This is explained at the end of ch.29 of Srednicki.

Asymptotically free theories like QCD are believed (though not yet proven) to exist in 3+1D.

 

[Gedankenspiel]

So QFTs on an infinitely extended lattice in 3+1 dimensions can be rigorously defined?

 

[Avodyne]

As a limit of a finite lattice, sure. Infinite extent isn't the problem, taking a limit where the lattice spacing goes to zero (in units of some physical length, like the Compton wavelength of the proton) and ending up with a Lorentz invariant interacting theory is the problem. The general belief is that this can be done in asymptotically free theories, but not in theories that are not asymptotically free. There is lots of evidence that non-asymptotically free theories cannot be constructed, e.g. http://arxiv.org/abs/0808.0082

 

[atyy]

At the informal level, there is a slight generalization of "asymptotically free" to "asymptotically safe". An asymptotically free theory is a type of asymptotically safe theory. Asymptotically safe theories are believed to have a good chance of being rigourously formulated.

Anyway, to start, one doesn't need to be rigourous. The most important physical idea is Wilson's idea that we don't take the cutoff to zero, and our best theory - the standard model of particle physics - is only valid at low energies. Wilson's great idea is in Srednicki's Chapter 29 "Effective Field theory". After Wilson, the non-rigrouous physicists believe "we understand quantum field theory" in a way that Feynman and Dirac did not, because in their day renormalization was the mathematically nonsensical procedure of subtracting infinities.

An even more beginner's point is that non-relativistic quantum mechanics for many identical particles can be exactly reformulated in the language of non-relativistic QFT (for historical reasons, this reformulation is called "second quantization" although that is a misnomer since it is exactly equivalent to non-relativistic QM of many identical particles which is already quantized). Then relativistic QFT uses the same language but with the possibility of particle creation and destruction and relativistic symmetry. This is mentioned at the start of Srednicki's book. Another useful reference is http://hitoshi.berkeley.edu/221B/QFT.pdf - see the part on the quantized Schroedinger field.

Free draft version of Srednicki: http://web.physics.ucsb.edu/~mark/qft.html
Free notes by Schwartz who also has a QFT book: http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf

From here one sees the greatness of the ancients - how on Earth did anyone learn QFT from books like Bjorken and Drell? Well, I guess they didn't. They were blundering around in the dark with great good luck, until Wilson came along.

 

[Demystifier]

I am looking for a detailed treatment of the foundations of QFT starting at non-relativistic QM.

If you are interested in conceptual foundations of QFT from a slightly philosophical point of view, I recommend
https://www.amazon.com/dp/0691016275/?tag=pfamazon01-20
At the beginning it briefly mentions some basics of non-relativistic QM, but emphasis is on QFT.

For a review of some foundational conceptual problems in non-relativistic QM, relativistic QM, and QFT see also
http://lanl.arxiv.org/abs/quant-ph/0609163

 

[Gedankenspiel]

@all: Thanks for the loads of literature. I'll try my best to find the answers to my question there.

@Avodyne: As I understand quantum field theory, literally, it is a quantum theory of fields, so it deals with functionals of fields. To define a useful scalar product in this space of functionals, we need to integrate over the space of field configurations, which is infinite dimensional, even if we define the theory on a lattice space-time. Do we have a suitable measure to perform such an integration? You seem to imply that in the limit of finite lattices we do. Can you tell more about that?

@atyy: I agree that the Wilsonian understanding of renormalization has led to valuable insights into QFT. You say the right thing: to start, one doesn't need to be rigorous. Neither theories do nor a learner of the theory does. But after decades of QFT the call for rigour doesn't seem to be unjustified to me. But it seems to me that it was louder in the 50s and 60s than it is today.

 

[Avodyne]

On a spatial lattice, a scalar field is just a set of harmonic oscillators. (Gauge fields are a little more complicated, but no harder to rigorously define.) There is no issue with inner products or measures on a lattice.

The Wilsonian point of view (which is deeply tied to lattice regularization) has provided almost all the insight we currently have about QFT in 3+1D. The attempts at rigor of the 50s and 60s (which were based on continuous fields and Lorentz invariance) continue to this day, but IMO have failed to do much of anything useful.

 

[George Jones]

General relativity does not start at Newtonian gravity because it introduces a completely new concept. In contrast, relativistic quantum theory is based on the same principles of quantum theory as non-relativistic quantum theory is. So I wonder if it is possible to gradually extend well understood theories to finally end up with a relativistic quantum theory.

I think we disagree here. I think that sometimes conceptual leaps are necessary, and that the bridge between the framework with larger domain of applicability and the framework with smaller domain of applicability can be established only after a conceptual leap.

And if not, where exactly the problems arise. With "difficulty" I mean the fact that, as far as I know, no rigorously defined relativistic quantum theory in 3+1 dimensions is known. But exactly why?

Okay, I misinterpreted your first couple of posts, so I don't think that the reference I gave above will be very useful to you.

I short, nice book is "Quantum Field Theory: A Tourist Guide for Mathematicians" by Gerald Folland

https://www.amazon.com/dp/0821847058/?tag=pfamazon01-20

Although Folland doesn't cover as much as physics texts such as Schwartz or Peskin and Schroeder, Folland does cover a lot more than most rigourous math books on quantum field theory. Folland uses mathematical rigour where possible, and where physicists' quantum field theory calculations have yet to be made mathematically rigourous, Folland states the mathematical difficulties, and then formally pushes through the physicists' calculations.

I've been waiting for many years for someone to write this book, and now I don't have time to read it.

A thick book in which you might be interested is "The Conceptual Framework of Quantum Field Theory" by Duncan

https://www.amazon.com/dp/0199573263/?tag=pfamazon01-20

Duncan comments on some of the mathematical problems, e.g., Haag's theorem.

A series of thick books by Zeidler, "Quantum Field Theory: A Bridge between Mathematicians and Physicists", might address some of your questions. The first three volumes in the Series have been published, with more to follow,

https://www.amazon.com/dp/3540347623/?tag=pfamazon01-20
https://www.amazon.com/dp/3540853766/?tag=pfamazon01-20
https://www.amazon.com/dp/3642224202/?tag=pfamazon01-20

 

[dextercioby]

George, I have been waiting for a 4th volume of Prof. Zeidler series since 2011. I remember the plan of Reed & Simon to write a lot of volumes. They stopped after 4. Let's hope it won't be the case for these encyclopedic writings.

 

[Gedankenspiel]

I think we disagree here. I think that sometimes conceptual leaps are necessary, and that the bridge between the framework with larger domain of applicability and the framework with smaller domain of applicability can be established only after a conceptual leap.

I don't say that there might be no need for a conceptual leap. I was just wondering if QFT is based on one or not. It seems to me that is "just" bringing together two preestablished concepts: special relativity and quantum theory.

Probably there is a conceptual leap after all, but I just don't know what it is.

 

[Gedankenspiel]

On a spatial lattice, a scalar field is just a set of harmonic oscillators. (Gauge fields are a little more complicated, but no harder to rigorously define.) There is no issue with inner products or measures on a lattice.

But doesn't this bring us an issue with breaking Lorentz invariance instead? Can we define the theory on a lattice right from the start or do we have to resort to the continuum for having a Lorentz invariant theory, which we then regularize by using a lattice?

And by the way: is there any literature where this is done from scratch? I am less interested in numerical lattice simulations or the path integral formulation than in the canonical quantization formalism on the lattice.

 

[atyy]

@atyy: I agree that the Wilsonian understanding of renormalization has led to valuable insights into QFT. You say the right thing: to start, one doesn't need to be rigorous. Neither theories do nor a learner of the theory does. But after decades of QFT the call for rigour doesn't seem to be unjustified to me. But it seems to me that it was louder in the 50s and 60s than it is today.

Yes, but maybe one should make the Wilsonian viewpoint rigourous, not the idea that one needs exact Poincare invariance.

As an example, one can have a relativistic theory emerge from a non-relativistic theory - the massless Dirac fermions in graphene are an example of that. So at the fundamental level, we have a perfectly well-defined theory - a non-relativistic theory in finite volume (if you believe Copenhagen, you don't need infinite volume).

If you are interested in rigourous relativistic QFT, Vincent Rivasseau's book http://rivasseau.com/resources/book.pdf may be useful, and also some knowledge of the Osterwalder-Schrader conditions http://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem, which is one way of making the path integral for relativistic QFT rigourous.

I think one obstacle to the Wilsonian viewpoint is that so far there is no lattice standard model - the difficulty seems to be the chiral fermions interacting with non-Abelian gauge bosons.

 

[muppet]

I don't say that there might be no need for a conceptual leap. I was just wondering if QFT is based on one or not. It seems to me that is "just" bringing together two preestablished concepts: special relativity and quantum theory.

Probably there is a conceptual leap after all, but I just don't know what it is.

The standard answer to this, which is treated in many standard textbooks or widely available lecture notes, is that yes: there is a conceptual leap, which lies in the fact that relativistic quantum mechanics for fixed number of particles described by a conserved probability density is not consistent.

An alternative starting point might be to say that we take the existence of classical fields as a given, and try to apply the general principles of quantum theory to them. To a certain extent, this is usually the first thing textbooks do after elaborating on why SR +QM requires the field viewpoint.

 

[Doug137]

The question that began this thread -
Literature for QM to QFT step-by-step
is beautifully addressed in Bob Klauber's Student Friendly Quantum Field Theory. He shows the development of and compares the theories of Nonrelativistic QM, Relativistic QM and QFT. He also includes Nonrelativistic QFT but not much because it isn't taught much but logically he shows how it fits in. To say that Klauber is clear is a gross understatement. His book is very different from all the others primarily because he addresses the student new to the subject. He is not concise - he explains everything. His derivations can be followed without head scratching about "how did he get from here to there?" His scope is more limited than most QFT books simply because he covers the basic material so thoroughly. Good luck!
PS I am so impressed by his approach that I am applying some aspects of it to the development of a primer to Lie Algebras for Physicists because I have found the texts difficult to approach (like QFT texts!) and hope to assist others in getting started to learning Lie Algebras..

 

[micromass]

George, I have been waiting for a 4th volume of Prof. Zeidler series since 2011. I remember the plan of Reed & Simon to write a lot of volumes. They stopped after 4. Let's hope it won't be the case for these encyclopedic writings.

Simon will very soon publish 5 analysis books covering a lot of real analysis, complex analysis, harmonic analysis and operator analysis. It won't really continue Reed & Simon, but you might be interested.

 

[Gedankenspiel]

On a spatial lattice, a scalar field is just a set of harmonic oscillators. (Gauge fields are a little more complicated, but no harder to rigorously define.) There is no issue with inner products or measures on a lattice.

Let me take this up again. A scalar field on a spatial, infinitely extended lattice means that there is a one-dimensional configuration space at each lattice point (the field amplitude). But there are infinitely many of them, so the configuration space of such a field is infinite-dimensional. A wave function on this configuration space is a map from this space to the complex numbers. A scalar product in such a space of wave functions would be something like an integration w.r.t. an infinite product measure of the Lebesgue measure on the real line.

So it still seems to me that there is an issue with scalar products in wave function spaces on infinite-dimensional configuration spaces.

 

[Avodyne]

I'm not sure what your issue is. The infinite case is defined as the limit of the finite case. In principle, the limit could fail to exist, but this would have to manifest itself as some sort of pathological volume dependence in the finite case. It's hard to see how this could arise.

 

[Gedankenspiel]

I am asking because one often hears that path integrals cannot be made rigorous because there is no measure on field space. Field space on a lattice may be more benign because it has countable dimension but it is infinite nevertheless. As there is no analogue to the Lebesgue measure on an infinite dimensional space I am wondering how an integration can be rigorously defined.

Can you recommend literature where canonical quantisation is done on the lattice? Most books on lattice field theory quickly proceed to or even start out with the Lagrangian formulation.

 

[vanhees71]

Well, the problem is that also the operator formalism is not rigorous. Also you cannot do "canonical quantization" without the Hamiltonian formulation of Hamilton's principle, because it's based on the heuristic "reinterpretation" of Poisson brackets in the Hamiltonian canonical formulation as the (equal time) canonical commutation relations of field operators. If you put the system on a discrete finite lattice, you end up with a finite set of quantum degrees of freedom, which is of course rigorously definable, but the problem is the continuum limit.

 

[atyy]

I haven't read, but DuckDuckGo and Google gave:
http://scitation.aip.org/content/aip/journal/jmp/5/8/10.1063/1.1704212
http://projecteuclid.org/download/pdf_1/euclid.cmp/1103839938

 

[Gedankenspiel]

@atyy: Thanks a lot! I'll have a look at these.

@vanhees71: I have no problem with the heuristic deduction of quantum theory. My problem is if the resulting theory can somehow be rigorously defined. I also don't care for the continuum limit as the theory is defined on a finite length scale. But I wonder if we can have a proper definition on an infinitely extended lattice. Or is it pointless and should we confine ourselves to finite volume and finite lattice spacing?

Another point I still don't get: if the ground state of an interacting theory is a mixture of the ground state of the non-interacting theory (where all field mode occupation numbers are 0) and states with occupation number >0: why then don't we observe particles all over the place, just because they are part of the interacting vacuum?
If one tries to escape from this conclusion by saying that the notion of a particle in the interacting theory is different from the notion in the non-interacting theory: how is a particle in the interacting theory defined then? And if it is defined as a state with E² = m² + p², how is it guaranteed that we still have states which fullfil this relation, given the fact that energy and momentum operators change in the interacting theory?

Is there any literature which covers these kind of questions?

 

[atyy]

If one tries to escape from this conclusion by saying that the notion of a particle in the interacting theory is different from the notion in the non-interacting theory: how is a particle in the interacting theory defined then? And if it is defined as a state with E² = m² + p², how is it guaranteed that we still have states which fullfil this relation, given the fact that energy and momentum operators change in the interacting theory?

Is there any literature which covers these kind of questions?

The usual way to do is just ignore all the difficulties and use the "interaction picture". There is a famous theorem called Haag's theorem that shows that the usual proof is wrong. Fortunately, the wrong proof gives the right formula, and rigourous quantum field theorists can derive the same formula by other means (at least that's what I understand). This is discussed in the book by Rivasseau mentioned in a previous post.

However, there are good physical reasons to use the wrong-in-infinite-volume derivation - which is that we should not expect our theory to fail in large but finite volume.