18
votes
Accepted
QED and anomaly
1. How can we show that $\partial\cdot j\equiv 0$ at the quantum level?
For example, by showing that the Ward Identity holds. It should be more or less clear that the WI holds if and only if $\partial\...
15
votes
How are anomalies possible?
It's true as the others have said that the path integral measure may not be invariant.
However I don't really like that quote as a general description of anomalies, since it sounds like they rely on a ...
- 8,134
13
votes
Accepted
Why is there no anomaly when particle mechanics is quantized?
Quantum mechanics can also become anomalous. An example is a charged particle moving in a uniform magnetic field. On the classical level, the system is translation invariant in both x- and y-direction....
- 12k
13
votes
Accepted
How to calculate an axial anomaly in 1+1 dimensions?
The axial anomaly.
In this answer we calculate the axial anomaly, following Fujikawa's method, in an arbitrary number of spacetime dimensions $d$. We assume $d\in 2\mathbb N$ inasmuch as there is no $...
12
votes
Accepted
Anomaly is due to the noninvariance of the path-integral under a symmetry. Is the noninvariance reflected on 1PI effective action?
The effective action $\Gamma[\phi_c]$ is the Legendre transform of the generator of connected correlation functions $W[J]$ which for most QFTs is defined as
$$W[J]=-i\log Z[J]$$
$Z[J]$ is the ...
- 2,223
10
votes
Anomalies in QFT
Anomalies is very popular and fast growing theme, so I will try collect main terminology and give major references.
Anomalies with chiral fermions:
David Tong: Lectures on Gauge Theory
Jeffrey A. ...
10
votes
Accepted
Anomaly, symmetries, and Ward identity
Background
Here we work in the Euclidean theory throughout. I also preface this with a disclaimer that I have been a bit lax with indices, but hopefully the message remains clear.
The Ward identities ...
- 8,570
10
votes
Accepted
How projective representations can lead to 't Hooft anomalies in quantum mechanics?
Shu-Heng Shao's argument is essentially the statement that gauging means projecting to $G$-invariant subspace in (0+1)d. In a way this is the definition of gauging in (0+1)d in the Hamiltonian ...
- 7,149
9
votes
Accepted
Can anomalies exist without gauge fields?
I would defer to the extant answer, only to adapt its item 2 to your potentially narrower question. Matt S is somewhat glib here, in that there would be something in theories without [manifest, overt] ...
- 66.3k
9
votes
Accepted
When is an anomaly one-loop exact?
Before answering the question, let me remark that number of loops required to obtain the anomaly exactly isn't an invariant quantity; the anomaly is physical, but the number of loops is not.
In the ...
- 31.2k
9
votes
Accepted
Why do we solve the Wess-Zumino consistency condition using the method of descent?
The case of perturbative anomalies is straightforward: the only such anomalies are of the ABJ type, i.e., they appear in even dimensions and are associated to chiral fermions. These are completely ...
9
votes
How are anomalies possible?
This is such a short answer I'm tempted to just leave it as a comment, but besides specifying the Lagrangian you need to also specify your regularization, and the regularization is what introduces the ...
- 8,917
8
votes
Accepted
The non-abelian chiral anomaly and one-loop diagrams higher than the triangle one
The higher polygonal diagrams do not contribute to the anomaly. If you examine the actual computation of the higher polygonal diagrams, you will find that they, due to their lower/non-existent ...
- 129k
8
votes
Accepted
Why is 2 a pseudoreal representation and there is no 2-2-2 anomaly?
Since isomorphism classes of irreducible representations of $\mathrm{SU}(2)$ are classified fully by their half-integer spin $j\in\frac{1}{2}\mathbb{Z}$, there is precisely one irreducible ...
- 129k
8
votes
Accepted
Are anomalies a failure of the canonical quantization prescription? Why not?
The replacement of the Poisson brackets by commutators per se is not yet a quantization. A quantization must include a prescription of how to map functions on the phase space into operators on the ...
- 31.2k
8
votes
How are anomalies possible?
The reason for this is that in the classical theory the Lagrangian fully specifies the dynamics of the system, however, in the Quantum system this is not true. Rather the Quantum theory is given by (...
- 1,199
7
votes
Why are there specifically 10, 11, or 26 dimensions in string theory?
In bosonic string theory, after canonical quantisation, we can construct particle states from the creation operators $\tilde{\alpha}^i_{-1}$ and $\alpha^j_{-1}$, as $\tilde{\alpha}^i_{-1} \alpha^j_{-...
- 19.5k
7
votes
Understanding typical non-perturbative calculations in QFT
1) The observables in field theory are (T-ordered) correlation functions. These correlation functions have perturbative (P) and non-perturbative (NP) contributions, but the relationship between the ...
- 19k
7
votes
Why are topological properties described by surface terms?
I am giving you an answer from the path integral quantization point of view.
When we quantize a gauge theory, we need to sum over all the configurations of connections of a gauge group on some ...
- 31.2k
7
votes
Accepted
Weyl anomaly in 2d CFT (string theory lectures by D.Tong)
TL;DR. The main point is that Tong only needs to identify the leading singularity in order to determine the Weyl anomaly
$$ \langle T^{\alpha}{}_{\alpha}\rangle~=~-\frac{c}{12} R^{(2)}. \tag{4.35} $$...
- 213k
7
votes
Accepted
How do anomalies affect the field equations of motion?
Short answer: The operator $\hat j$ is meaningless as written, because it contains divergences. In order to manipulate this operator (and take its divergence) one must introduce a regulator which, ...
7
votes
Why does a triangle anomaly appear in a gauge theory?
Perspective
We don't assume that there should be an anomaly. For a given pattern of interactions between gauge fields and matter fields, the quantum theory either does or does not have a chiral ...
- 55k
7
votes
Accepted
Time-reversal (explicitly) broken surface of $(3+1)$-dimensional topological insulator
Great question. The answer is no, the surface is not anomalous if $\mathbb{Z}_2^T$ is explicitly broken. The reason is that the Hall conductance of the surface (which allegedly is 1/2) is not actually ...
- 3,506
7
votes
Accepted
Why do we think that the $U(1)$ problem is solved by instantons?
I agree that it is not so much instantons that are relevent, but the anomaly and $\theta_{QCD}$ term. By the early 1970's, when I became a grad student, the anomaly was well understood to be the ...
- 56.6k
7
votes
How can we know a gauge theory is not anomalous?
$\newcommand{\D}{\mathrm{D}} \newcommand{\d}{\mathrm{d}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\Ds}{D\kern-.6em/\kern.1em} \newcommand{\B}{\mathrm{B}} \newcommand{\H}{\mathrm{H}} \newcommand{\SU}{\...
- 4,842
7
votes
Accepted
How is the pion related to spontaneous symmetry breaking in QCD?
In a world without EW SSB, pions would, indeed, be perfect massless! Pion masses reflect two different SSBs.
There are two SSB scales involved in the SM: the electroweak spontaneous breaking of $SU(2)...
- 66.3k
6
votes
Why is there no anomaly when particle mechanics is quantized?
Anomalies are not particular to quantum field theory, or even to quantum theory. An anomaly is an obstruction to representing some physically relevant group/algebra, often a symmetry group or an ...
- 129k
6
votes
Accepted
How is Berry phase connected with chiral anomaly?
I have been collaborating with the authors of that paper. In Section 2.6 of my thesis I explained the relations between the full quantum computation of chiral anomaly, the (semi-classical) Nielsen-...
- 356
6
votes
Conformal theory with zero central charge
Here an easy way to think about your problem in QFTs (EDIT: I thank ACuriousMind for having pointed out that my argument does not hold in string theory)
Is c=0 meaningful? Consider the case $d=2$.
...
- 2,237
6
votes
Accepted
Particle on a circle with magnetic flux$.$
The Lagrangian density is supposed to be a $\mathbb{R}$-valued function. As long as it only includes derivatives of functions on the circle, it is straightforwardly $\mathbb{R}$-valued because the ...
- 129k
Only top scored, non community-wiki answers of a minimum length are eligible