Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

All Questions

1
vote
0answers
23 views

Implications of Instanton Corrections (to Degenerate Vacuua) for Spontaneous Symmetry Breaking

We consider that if the classical vacuua of a theory are degenerate then each of them can be non-invariant under one or more of the symmetries of the Lagrangian. We can choose one of the vacuua and ...
6
votes
0answers
111 views

Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields $$ \langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\...
1
vote
0answers
48 views

Nonperturbative results for $\phi^3$ theory in dimensions $d>6$?

The theory is nonrenormalizeable in those dimensions, but can you say anything about the theory anyway? Specifically I am wondering about the status of whether the theory is trivial, i.e. a ...
2
votes
1answer
67 views

Non-renormalizeable Interaction Implies Trivial Interaction?

It has been rigorously proved that the $\phi^4$ theory is trivial, i.e. is a generalized free field, in spacetime dimensions $d>4$. It is also the case that this theory is non-renormalizeable in ...
4
votes
1answer
80 views

Nonrelativistic Quantum Mechanics Results Implying Analogous QFT Results?

One particularly fascinating example of this I have found is the following. The delta function potential has no effect in nonrelativistic quantum mechanics in spatial dimensions greater than or equal ...
3
votes
2answers
129 views

Triviality of Yang Mills in $d>4$?

It has been proved that the $\phi^4$ theory is trivial in spacetime dimensions $d>4$. By trivial I mean that the field $\phi$ is a generalized free field, or in other words, it's only nonzero ...
0
votes
0answers
56 views

How does Atiyah-Singer index theorem relates instanton number to number of fermion zero modes?

I was studying this paper, where the authors consider an $SU(2)$ gauge field of instanton number 1 on a 4-sphere $M =S^4$. If $n_L$ is the number of zero modes of $\psi_L$ and $n_R$ is the number of ...
1
vote
1answer
60 views

S Duality and Effective Couplings

I am brand new to this subject, so this will probably be a very stupid question, but I would appreciate any patient explanations. S-duality is typically described as a relationship between two QFTs (...
22
votes
2answers
416 views

Can we get full non-perturbative information of interacting system by computing perturbation to all order?

As we know perturbative expansion of interacting QFT or QM is not a convergent series but an asymptotic series which generally is divergent. So we can't get arbitrary precision of an interacting ...
4
votes
1answer
108 views

Is there any proof that any result from perturbation theory is necessary an asymptotic series?

I know that almost all the series coming from perturbation theory are divergent, such as those from eigenvalue problems or the S-matrix in quantum field theory. The lore is that the series are ...
4
votes
0answers
110 views

What is the problem of non-pertubative quantisation?

In reading books about quantisation, there is (sometimes hidden) the claim, that quantisation is done using a pertubative approach. You look at the free field, find that it is essentially a sum of ...
5
votes
1answer
244 views

What's the difference between a gauge theory with group $G$ and one with its universal cover?

Consider a gauge theory with gauge group $G$, which is not simply connected. What is the difference between this theory, and one with gauge group $\tilde G$, the universal cover of $G$? Sharing the ...
4
votes
1answer
152 views

How to path-integrate over the half-line?

Consider the path-integral over a scalar field $\varphi$: $$ Z=\int_{\mathcal S}\ \mathrm e^{iS[\varphi]}\mathrm d\varphi $$ where $\mathcal S$ is some function space (say, Schwartz or its dual). How ...
14
votes
1answer
417 views

Why do we care about old-style, counterterm renormalizability?

There are a few different definitions of renormalizability that are standard in quantum field theory textbooks. They're all called the same thing, but I'll make up names to make the distinctions clear....
4
votes
2answers
341 views

What do we mean when we say 't Hooft proved that Standard Model is renormalizable?

This question is inspired from Why should the Standard Model be renormalizable? Ron Maimon says that standard model is renormalizable, and though there seems to be conflicting (?) answers. Is this ...
6
votes
0answers
106 views

Question about the vacua of the Standard Model

This question is probably based on a misunderstanding. Please correct me if I'm wrong, and if unclear, I'll try to put it in a clearer language. In Yang-Mills theory such as the theory of strong ...
11
votes
3answers
596 views

Books on non-perturbative phenomena in quantum field theory

I am looking for any good places (preferably textbooks) to study about introductory non-perturbative phenomena in Quantum field theory. Any suggestion will be appreciated.
3
votes
0answers
202 views

What is the current situation about triviality of $\phi^4$ theory in $d=3+1$?

I was reading a book by Franco Strocchi, this one, and in some points the author claims that the case of $d=3+1$ of triviality of $\phi^4$ theory is now proven. As far as I can tell, we have just some ...
22
votes
3answers
3k views

Does QED really break down at the Landau pole?

In QED, the fine structure constant $\alpha$ runs upwards in the UV, with a loop calculation (involving a geometric series of the vacuum polarisation diagram) indicating a divergence in $\alpha$ at $\...
3
votes
1answer
146 views

Baryon number violation in the Standard model at the perturbative and non-perturbative level

This is a continuation of my question here. Page 635 of this book by Matthew Schwartz effectively says that the $\partial_\mu J^\mu_B\neq 0$ where $J^\mu_B$ is the baryon current i.e., the baryon ...
3
votes
2answers
136 views

Is renormalizability only a problem in perturbation theory?

As far as I know, renormalization is needed when a scattering amplitude is divergent at some order of the coupling constant in a perturbation theory. So my question is whether the divergence (and thus ...
4
votes
2answers
526 views

Meaning of perturbative and non-perturbative renormalizability

What is meant by a theory to be (1) perturbatively renormalizable, (2) perturbatively non-renormalizable, (3) non-perturbatively renormalizable, and non-perturbatively non-renormalizable? In each case,...
4
votes
1answer
363 views

Why is the temporal gauge $A_0=0$ so popular in discussions of non-perturbative effects?

Almost every discussion of non-perturbative effects in Yang-Mills theory mentions in passing that they work in the temporal gauge. Why is this the case? A good example is the QCD vacuum. Almost ...
1
vote
1answer
115 views

Are all field interactions carried out through force-mediating particles?

To my knowledge, all field interactions are carried out through force-mediating particles. For example, electromagnetic interactions are carried out through exchanging photons. However, under the ...
7
votes
2answers
932 views

Understanding typical non-perturbative calculations in QFT [closed]

Perturbative calculations in quantum field theory are based on S-matrix expansion and calculating the Feynman diagrams. These Feynman diagrams are related to the scattering cross-sections and decay ...
12
votes
1answer
362 views

On the asymptotics of interacting correlation functions

Consider an interacting QFT (for example, in the context of the Wightman axioms). Let $G_2(x)$ be the two-point function of some field $\phi(x)$: $$ G_2(x)=\langle \phi(x)\phi(0)\rangle $$ Question: ...
3
votes
0answers
246 views

Perturbative coupling for QFT

I'm confused about the definition of a perturbative coupling for QFT that it should be less than 4 $\pi$, because the higher order corrections comes of order $\lambda/(4 \pi)$ .. Now why QCD is not ...
5
votes
0answers
195 views

Higher category theory, renormalization, and non-perturbative QFTs

I'm (vaguely) aware of certain uses of higher category theory in attempts to mathematically understand quantum field theories -- for example, Lurie's work on eTQFTs, the recent-ish book by Paugam, and ...
4
votes
0answers
264 views

Non-perturbative approach to QFT in Hamiltonian formalism?

A simple conceptual question today: is it true that QFT can only be approached in a non-perturbative way only through the functional methods (like 1/N), while in the Hamiltonian formalism we can only ...
9
votes
4answers
486 views

Mass Renormalization: Geometric Series of One Particle Irreducible Diagrams

Pretty much everywhere I look it is stated that the full two point Green function (let's say for the Klein-Gordon field) is a geometric series in the one particle irreducible diagrams, ie. in momentum ...
4
votes
1answer
98 views

Why do we have to sum the expansions around all the action's stationary points?

This is in some sense a follow-up question to my previous question Why is it OK to keep the quadratic term in the small $\hbar$ approximation?. I understand how we can expand the action around a ...
4
votes
1answer
96 views

Experimental observation of non-perturbative effects

Many quantum field theories come with non-perturbative objects such as solitons and instantons, and non-perturbative effects such as the Schwinger effect. However, it is hard to find any review on ...
4
votes
0answers
194 views

Some questions about QCD [closed]

About QCD, I have two questions. I know I should propose one question one time, but they are actually two steps of the same question: Non-perturbative aspects of QCD. 1, Why do we need to solve QCD ...
3
votes
1answer
317 views

How do we calculate the S-matrix using non-perturbative QFT?

The cross section of a scattering process in $QFT$ is computed in terms of the S-matrix elements. In perturbative $QFT$, the same is done by computing the S-matrix elements by using Feynman ...
2
votes
1answer
1k views

Baryon number violation in the Standard Model

Anomaly cancellation in the Standard model requires $B-L$ to be constant, which is done using perturbative diagrammatic expansion. Secondly, baryon number is conserved as an $U(1)$ global field ...
4
votes
0answers
75 views

Topological terms VEVs and ghosts

Suppose we have the Standard model, and we want to calculate with VEVs of topological susceptibilities of $SU_{L}(2), U_{Y}(1)$ and $SU_{c}(3)$ fields, which have the form $$ \tag 1 \kappa \equiv \...
6
votes
2answers
1k views

What are non-perturbative effects and how do we handle them?

Schwartz's QFT book contains the following passage. To be precise, total derivatives do not contribute to matrix elements in perturbation theory. The term $$\epsilon^{\mu\nu\alpha\beta} F_{\mu\...
0
votes
0answers
652 views

Physical meaning of Ward Identity and computing vertex functions

Following the derivation of Ward Identity by Weinberg book, you get it in the form $$ (l-k)_\mu S'(k)\Gamma^\mu(k,l)S'(l) = i S'(l) - iS'(k) $$ Can anyone explain the physical meaning of this ...
4
votes
1answer
249 views

Mathematical proof that $\exp(-1/|g|)$ is always related with formation of bound states through scales?

I know that this function ($g$ means coupling) is non-analytical in $g=0$, so this function is only appreciable under non-perturbative calculations, so is a non-perturbative phenomena. This function ...
6
votes
1answer
509 views

How do instantons look in real time/spacetime?

Instantons, as I understand it, are mathematical constructions in Euclidean spacetime. Does it imply that instantons do not exist in real spacetime or the instanton tunneling effects does not have ...
1
vote
1answer
59 views

Translational versus dilatational zero modes?

Why are the zero modes of the SU(2) Yang Mills instanton referred to as translational or dilatational zero modes? Is this standard terminology?
2
votes
1answer
1k views

difference between classical vacuum solutions and instantons

What does the classical vacuum of the $SU(2)$ Yang-Mills action correspond to? Does it correspond to $F_{\mu\nu}=0$ everywhere or just at the spatial infinity? In Srednicki’s book, he has shown that, ...
8
votes
1answer
352 views

Do we need new physics to supercede triviality?

I've been reading about the higgs triviality bound (see for example here). It is discussed that the higgs self coupling at some energy scale becomes non-perturbative. If the higg's mass is above about ...
14
votes
2answers
812 views

Non-Perturbative Feynman diagrams?

The Wikipedia page for Feynman Diagrams claims that Thinking of Feynman diagrams as a perturbation series, nonperturbative effects like tunnelling do not show up, because any effect that goes to ...
3
votes
1answer
283 views

Significance of total divergence anomaly term

What is the significance of the fact that the anomany term (calculated from the triangle diagram) is a total divergence? Or, in other words, what is the significance of $$\partial_\mu j^\mu_A\sim Tr(W\...
2
votes
0answers
66 views

Theory with interaction and the birth of bound states during propagation

Suppose we want to calculate vacuum expectation $$ \tag 1 D_{lm}(x - y) = \langle \Omega | \hat {T}\left( \hat {\Psi}_{l}(x)\hat {\Psi}_{m}^{\dagger}(y)\right)| \Omega\rangle = \langle \Omega| \hat {...
1
vote
1answer
358 views

One more time about LSZ-theorem

This question is the continuation of this one. For simplicity, let's use $(1)$ from the linked question (it is called n-point Green function and in particle case coincides with internal diagram), $$ ...
6
votes
0answers
1k views

LSZ reduction theorem derivation in Weinberg QFT

When deriving LSZ reduction theorem Weinberg in his QFT book have assumed n-point generalized Green functions, $$ G(q_{1},...,q_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}q_{j}x_{j}} \...
2
votes
0answers
250 views

Definition of a non-pertubative Quantum field theory

How do you define a non-perturbative Quantum field theory. What does it mean? I was just digging around some math about the meaning of $Z_1$, and such in terms of probabilities. It turns out these and ...
4
votes
0answers
86 views

Mirrored decoupling fermion doublers and a lattice chiral fermion / gauge theory

Nielsen Ninomiya Fermion-doubling problem has known to be a challenge to construct a chiral fermion or chiral gauge theory on the lattice. There is a proposed resolution to use so-called two mirrored ...