# Noether vs Ward Takanashi

1. Oct 14, 2014

### the_pulp

This is what wikipedia says. Nevertheless, I dont think that it is true. I mean, the conservation of electric charge can follow from noether theorem field generalization applied to Electrodynamic Lagrangian, am I right? Is it wikipedia wrong by saying that conservation of electric charge can not follow from noether theorem? Or at least, not exactly right?

Thanks!

2. Oct 15, 2014

### dextercioby

Noether's theorem is valid only in classical field theory (which of course doesn't really exist for a Dirac field, for example, or even for a complex scalar field). <Conservation of electric charge> in the wiki context is meant in the sense of QFT, where you work with operators (operator valued distributions) and expectation values.

3. Oct 15, 2014

### the_pulp

Thanks for your answer. Nevertheless, in Ryder's "Quantum Field Theory", in page 90, where he talks about Complex Scalar Lagrangian, he derives a sort of electric current from the symmetry condition. This is a pure Classical Field Theory derivation that could be inferred by Noether theorem.
I know that life does not behave as Complex Scalar Lagrangian in Classical Field Theory says. What I try to be sure (or to rectify) is that in classical field theories, conservation of a sort of electric current can arise and that it is not a property of QFT through TW Identity (which seems to be in conflict with what wiki says). Am I right?

Thanks

4. Oct 15, 2014

### dextercioby

That's completely correct.

Yes, in the realm of classical dynamics (particles or fields), you only have the Noether theorem which provides you with the theory's conserved observables. They are then used in the so-called canonical quantization, so that one may be tempted to say that always classical -> quantum, when actually it's the other way around. But this is how the textbooks are written. They always start with the classical Lagrangians (which for the fields other than the e-m one or the gravitational one don't really exist), then bring in symmetries, Noether theorem, conserved charges then bam, canonical quantization. Makes you think that there's no quantum theory that didn't come from a classical counterpart through the 'Dirac trick'. But quantum theories can be built on their own, that's what axiomatical QFT for example does.

5. Oct 15, 2014

### samalkhaiat

What is wrong with Grassmann numbers? This is beauty of the Lagrangian formalism: the fields can take values in a completely arbitrary algebra.
Is there :) ? This is why he was a great man.
There has been a deep mathematical theory behind each one of “Dirac’s tricks”. His delta function paved the way for the Distribution Theory, and the “Poisson to commutator” trick led to the theory of Quantum Groups and quantization became a Lie algebra Deformation.
The program of the “Axiomatic QFT” is more than 60 years old yet, unfortunately, it failed to achieve its goal. The program produced countless theorems each with “weak and strong” version, but it can not construct interaction Hamiltonian for the simplest non-abelian multiplets. Ironically, the fact that Axiomatic QFT cannot account for Noether theorem is the main (if not the only) reason for its failure.

6. Oct 16, 2014

### dextercioby

Grassmann variables are a necessity stemming from quantum theories. They are mathematical artifacts for 'classical Lagrangians'.
Yes, by failing. :)
I offered this example just to show that one can think of QFT ouside its connection with the classical field theories.

7. Oct 16, 2014

### DarMM

There are quantum theories which are not the quantisation of a classical theory. They aren't too useful in relativistic field theory, although again, there are relativistic field theories which are not quantisations of classical ones, but just to say they do exist.

Axiomatic Field Theory as such wasn't concerned with constructing interacting field theories (that is Constructive Field Theory) rather it was concerned with seeing what mathematical system quantum field theories belong to (i.e. what sort of mathematical objects are fields) and then seeing what are the minimum set of assumptions on those objects in order for the work of physicists to be justified. It was then concerned with seeing what general theorems could be proved from those minimal assumptions.
Also interacting Hamiltonians have been constructed for non-Ableian gauge theories, but only in 1+1 and 2+1 dimensions.

Concerning Noether's theorem, I am not aware of an example of constructing a field theory that failed due to Noether's theorem, or how Noether's theorem prevented the construction of a field theory. Could you expand on this?

My understanding is that due to anomalies we wouldn't want Noether's theorem to be a fundamental part of QFT. Anomalies being examples of where application of Noether's theorem produces conserved currents, which are in fact not conserved in the Quantum Theory.
What prevents them from being conserved is in fact the fact that fields are distributions so that their products are ill defined, for example the chiral current in QCD, under Noether's theorem we have:

$\partial_{\mu}j_{5}^{\mu} = 0$

However since $j_{5}^{\mu}$ is a product of distributions, the fields themselves:

$j_{5}^{\mu} = \bar{\psi}^{a}\gamma^{\mu}\gamma^{5}\psi_{a}$

it is not well defined. We can define it in a mathematically rigorous way (extension of distributions, Epstein Glaser) or a physicists way (momentum space renormalisation), either way we get a new well-define version of $j_{5}^{\mu}$, let's call it $j_{5, R}^{\mu}$ however it then turns out that:

$\partial_{\mu}j_{5, R}^{\mu} \neq 0$

in fact:

$\partial_{\mu}j_{5, R}^{\mu} = \frac{N}{8\pi^2} F \wedge F$

which is responsible for the decay rate of pions and the heavy mass of the eta prime for example, very physical effects coming from the fact that Noether's theorem did not work.

EDIT: Personally I think anomalies are a good example to show that renormalisation is not some silly trick. That is, the fields are distributions, you can't just write down products without thinking and renormalisation is the physicists way of correcting these ill-defined products, i.e. working out the correct product and behold the correct product does not have the properties you'd expect, but its properties do show up in experiment.

Last edited: Oct 16, 2014
8. Oct 16, 2014

### DarMM

By the way, if anybody is wondering the real difficulty in constructing field theories in 3+1 dimensions is coupling constant renormalisation. Even in lower dimensions, theories with coupling constant renormalisation have never been constructed.

Originally, in the 1970s, there were problems with constructing theories that were only renormalisable as opposed to super-renormalisable. However this was successfully solved in the 1980s. Having the coupling constant be renormalised (by which I mean that for the physical coupling to be finite, the bare one must go to infinity or zero in the continuum limit) introduces a host of complications that has so far prevented analytic control over the theory.

Again I do not think that the Ward-Takahashi identities (quantum version of Noether's theorem) have posed any problem.

9. Oct 17, 2014

### samalkhaiat

Actually, I meant to criticize both AQFT and CQFT(after all, CQFT is nothing but synthesis of formal perturbative and axiomatic approaches).
In the Lagrangian formalism, thanks to Noether theorem, the generators of symmetry (including the Hamiltonian) are expressed in terms of fields. This luxury does not exist in the AQFT and CQFT.

See CH6.2 of J. Lopuszanski. “An Introduction to Symmetry and supersymmetry in QFT”, World Scientific.1991.
The departure of $\partial_{ a } J^{ a }_{ 5 }$ from its canonical value follows from: i) the intrinsic singularities of QFT, and ii) preserving gauge invariance. It is not Noether-theorem’s job to regulate those singularities for you.
In massless QED, the anomaly is often described as a “clash between two symmetries”: gauge invariance and axial symmetry. It is not possible to maintain both symmetries in perturbation theory, however, either one can be satisfied.
Due to the presence of a dimension 4 operator $\bar{F}^{ a b } F_{ a b }$ in the divergence, the renormalized axial current fails to generate chiral transformation. The corresponding axial charge is not conserved; it does not have the correct equal-time commutation relations with the renormalized fermions field.
In spite of the fact that conserved gauge-invariant axial current does not exist, axial symmetry can still be implemented in the theory. Indeed, it is possible to construct the following conserved bare current
$$\hat{ J }_{ 5 }^{ a } = J_{ 5 }^{ a } - \frac{ e^{ 2 } }{ 4 \pi^{ 2 } } \bar{ F }^{ a c } A_{ c } .$$
It can be shown that this operator is finite and does not need renormalization; the associated charge
$$Q_{ 5 } = \int d^{ 3 } x \hat{ J }_{ 5 }^{ 0 } ( x ) ,$$
is time-independent and generates the correct symmetry transformations for fermions and photons:
$$[ i Q_{ 5 } ( t ) , \psi ( t , \vec{ x } ) ] = \delta_{ 5 } \psi ( x ) ,$$
$$[ i Q_{ 5 } ( t ) , A ( t , \vec{ x } ) ] = 0 = \delta_{ 5 } A ( x ) .$$
Under the gauge transformations, $\delta \psi = i \Lambda \psi$, $\delta A_{ c } = \partial_{ c } \Lambda$, we find that $Q_{ 5 }$ is guage invariant:
$$\delta Q_{ 5 } = \frac{ - e^{ 2 } }{ 4 \pi^{ 2 } } \int d^{ 3 } x \ \partial_{ j } ( \Lambda \bar{ F }^{ 0 j } ) = 0 .$$
Therefore, chiral symmetry remains a good symmetry in spite of the anomaly in the divergence. This has the consequence that any property, based on axial symmetry, will be maintained in perturbation theory (the anomaly cannot be used to generate a mass term for the fermion).

10. Oct 17, 2014

### DarMM

It is? Most of constructive field theory uses very few ideas from perturbation theory. Rather it uses the scaled cluster (propogator) expansion and even then not as a perturbative expansion. Could you comment on how Constructive field theory uses perturbation theory, with reference to, let us say, the construction of $\phi^{4}_{3}$ as a simple example?

In the examples of the construction of a field theory that I am familiar with, the Langragian formalism along with symmetry generators written in terms of the field do appear. Could you explain a little, with reference to a paper involving an actual construction of a field theory in constructive field theory how does luxury does not exist?

I don't think it is Noether's theorem's "job" to regulate singularities, I'm actually not sure what that would mean, but it is still an example where the theorem fails. So it cannot be fundamental to quantum field theory. Instead the Ward-Takahashi identities are.
Renormalisation invalidates the conclusions of Noether's theorem in this case regardless of whether regularization is its job.

11. Oct 17, 2014

### atyy

Two questions regarding AQFT and CQFT.

1. Do any of the successful constructions by CQFT also fulfill AQFT axioms, so that the sucess of CQFT also means AQFT is succesful?

2. The Bisogano-Wichmann theorem does show up in modern non-rigourous QFT research, eg. http://arxiv.org/abs/1102.0440. Is this tangential to AQFT, or does it indicate that AQFT ideas are in fact used in non-rigourous research also?

12. Oct 17, 2014

### DarMM

Regarding (1.) all constructed quantum field theories satisfy the axioms of axiomatic quantum field theory. Even Yang-Mills in 3+1 has been shown by Balaban to obey some of the Wightman axioms, although we don't know yet if it obeys them all, we don't have enough control over the continuum limit.

By the way, for gauge field theories it is the observable fields (I'm not sure what people prefer to call them, I mean the gauge invariant fields) that obey the Wightman axioms, not the $A_{\mu}$ gauge potential.

13. Oct 17, 2014

### DarMM

Regarding your second point, I would say that AQFT ideas appear a bit in non-rigorous research (such as the Källén–Lehmann
representation of the propagator). However overall I would say it is tangential to mainstream research.

Personal comment: However I will say, based on papers I have read, previous posts on this forum and the stackexchange site, that people have odd ideas about what AQFT or CQFT are concerned with, as if the were competing with or an alternative to normal quantum field theory. They're really nothing more than areas concerned with showing that what physicists do everyday is mathematically valid, that is all.
Think of it like how in Reed and Simons they actually prove that the Hydrogen atom is self-adjoint, where as a normal textbook will just show it is symmetric. Nothing wrong with either way of doing things. Getting too bogged down in mathematics can slow down physical thinking, but usually somebody will come along and mathematically tidy things up.
A perfect example is the existence of QCD. Now I have no doubt QCD exists mathematically, but I think mathematics should advanced to the point where this can be proven. The fact that we cannot prove that QCD and other 3+1-d theories actually exist due to being unable to handle coupling constant renormalisation indicates that our control of C*-algebra nets in the Algebraic/Hamiltonian view or more abstract stochastic processes (Yang-Mills theory in the path-integral approach is essentially a theory of random fiber bundles) in the path integral view, is not yet mature.

14. Oct 17, 2014

### atyy

I think there are things in classical statistical mechanics that even non-rigourous people find bizarre. The $\epsilon$-expansion and the multiplication of distributions in the KPZ equation. I think replica symmetry breaking used to be another one, but I hear that rigourous folks can handle that nowadays. Any progress on the other two? I hear Hairer's got KPZ, which I guess leaves $\epsilon$-expansion?

BTW, I remember hearing the story that Balaban stopped working on Yang-Mills because he moved house, and the moving company lost a box of his Yang-Mills work. Not sure where I heard that now though.

Last edited: Oct 17, 2014
15. Oct 17, 2014

### dextercioby

I don't find the exact reference in R&S's 4 volumes, but I believe a rigorous proof can be constructed via Kato's results in the 1950's. I could find a proof by theorem 10.2 (p. 219) in G. Teschl's <Mathematical Methods in Quantum Mechanics. With Applications to Schrodinger Operators> and his comment following on page 220.
Having futher checked, one can find the full details in Triebel's <Höhere Analysis> (1972, German) or <Higher Analysis> (1992, English), Chapter 7.

Last edited: Oct 17, 2014
16. Oct 17, 2014

### zoki85

17. Oct 17, 2014

### atyy

The story about Balaban's notes is told by Jaffe just after 1:15:00 in this talk:

Jaffe's talk was part of a workshop on the Mathematical Foundations of Quantum Field Theory at the Simons Center at Stony Brook, highlighted by Peter Woit in http://www.math.columbia.edu/~woit/wordpress/?p=4396.

18. Oct 18, 2014

### DarMM

It's in volume II, theorem X.15. It's part of their presentation of Kato's material, as you mentioned. I find it a nice proof of the self-adjointess. Thank you for your references, it's always nice to see how others explain things.

19. Oct 18, 2014

### DarMM

Interesting talk! I'm actually shocked that that is what happened!

20. Oct 18, 2014

### dextercioby

Just read this: http://www.arthurjaffe.com/Assets/pdf/CQFT.pdf. I found it useful.

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