Quantum Field Theory (QFT) is the theoretical framework describing the quantisation of classical fields which allows a Lorentz-invariant formulation of quantum mechanics. QFT is used both in high energy physics as well as condensed matter physics and closely related to statistical field theory. Use ...

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geometric quantization of the moduli space of abelian Chern-Simons theory

I wish to understand the statement in this paper more precisely: (1). Any 3d Topological quantum field theories(TQFT) associates an inner-product vector space $H_{\Sigma}$ to a Riemann surface ...
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466 views

Path integral and geometric quantization

I was wondering how one obtains geometric quantization from a path integral. It's often assumed that something like this is possible, for example, when working with Chern-Simons theory, but rarely ...
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340 views

Understanding Cherns-Simons-Witten Theory

I want to read about Wittens work, on Cherns-Simons theory, and relations to knots and jones polynomials. I am extremely motivated to read his paper: Quantum Field Theory and Jones polynomial. What ...
6
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196 views

Mass generation by Chern-Simons theory

Why the mass generation via a Higgs mechanism is different from that of Chern-Simons theory? I haven't done any formal course in Quantum field theory,so how do I understand this just having some basic ...
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3answers
560 views

The Chern-Simons/WZW correspondence

Can someone tell me a reference which proves this? - as to how does the bulk partition function of Chern-Simons' theory get completely determined by the WZW theory (its conformal blocks) on its ...
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391 views

Yang-Mills CP violation

Why does a term proportional to $\left(F,\,\tilde{F}\right)\propto Tr\left[ F_{\mu\nu}\tilde{F}^{\mu\nu}\right]$ in the Lagrangian of the pure Yang-Mills theory violate CP?
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38 views

Noncommutative Field Quantization

I'm studying noncommutative (quantum) field theory, and I have confusion need to be clear. I'm reading Szabo's and Douglas's .pdf of noncommutative QFT. As I understand, in the book they just ...
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2answers
104 views

What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
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1answer
114 views

Where does this delta of zero come from?

It is common when evaluating the partition function for a $O(N)$ non-linear sigma model to enforce the confinement to the $N$-sphere with a delta functional, so that $$ Z ~=~ \int d[\pi] d[\sigma] ~ ...
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1answer
63 views

How non-abelian gauge coupling runs below confinement or QCD scale?

I know the famous beta function of asymptotic free, but that seems describe the running coupling beyond confinement/QCD scale so that a perturbative analysis can apply. But how coupling runs below ...
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74 views

I want to decompose a tensor product using Littlewood-Richardson rule, How do I find the component of this in each irreducible space?

Let me set up the notation I am using. $(abc,de)$ denotes the standard Young tableau where the first row is $abc$ and the second row is $de$. Each young tableau corresponds to the young symmetriser, ...
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613 views

Why is the functional integral of a functional derivative zero?

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e. ...
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44 views

S-Matrix Generating Functional (Problem 4.1 in Weinberg)

I'm currently working through Weinberg's QFT book, but I'm somewhat stuck at problem 4.1, which states: Define generating functionals for the S-matrix and its connected part: \begin{equation} ...
7
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1answer
162 views

Coleman-Weinberg potential: resum at 2 loops?

Say we want to compute the Coleman-Weinberg potential at 2 loops. The general strategy as we know is to expand the field $\phi$ around some background classical field $\phi \rightarrow \phi_b + ...
5
votes
1answer
568 views

Analytic continuation of imaginary time Greens function in the time domain

Consider the imaginary time Greens function of a fermion field $\Psi(x,τ)$ at zero temperature $$ G^τ = -\langle \theta(τ)\Psi(x,τ)\Psi^\dagger(0,0) - \theta(-τ)\Psi^\dagger(0,0)\Psi(x,τ) \rangle $$ ...
2
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1answer
43 views

Fierz identity for Weyl spinors in tensor currents

Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ ...
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532 views

Hilbert Space of (quantum) Gauge theory

Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down the Hilbert Space of quantum Gauge theory with spin-S. (Are there something ...
5
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1answer
167 views

Secondary constraints leads to the value of lagrange multiplier

From Lagrangian I got two primary constraint $\phi_i$ and $\phi$. And my Hamiltonian in presence of the constraints becomes- $$H_p=p\dot q-L+\lambda_i\phi_i+\lambda\phi$$ here the $\lambda_i$ and ...
4
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539 views

Two photons transition

if an atom in its ground state is coupled to an electromagnetic field it can absorb a photon if the EM field contains one with the right frequency. These transitions depends on $⟨f|H_i|i⟩$ (from ...
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1k views

If subatomic particles pop into existence all the time, why don't I gain weight?

Watching Discovery's first episode of the first season of Curiosity (entitled "Did God Create the Universe?" by Stephen Hawking), I heard this information: [...] you enter a world where conjuring ...
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1answer
279 views

Labelling representations using isospin and hypercharge

Can someone explain how isospin and hypercharge can be used to label representations? What is the meaning of the term singlet, doublet etc in this context? In particular how can I use it to label ...
6
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122 views

The problem in Sredniki's textbook: How do I calculate loop corrections for $\phi\phi\to\phi\phi$ with this Lagrangian?

The problem in Sredniki's textbook 10.5 : For a free scalar field $\psi$, the Lagrangian is $$\cal{L}= -\frac{1}{2}\partial^\mu\psi\partial_\mu\psi-\frac{1}{2}m^2\psi^2$$ Here we use the metric ...
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49 views

About Shor's error correcting algorithm

In this paper, http://arxiv.org/abs/1301.4504 in equation 4.1 in what sense are the two states a "9-qubit state"? I did not understand this counting. Can someone explain what are the different $X_i$ ...
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92 views

Nature as a calculator [closed]

I wondered whether physical experiments can find the outcome of otherwise intractable mathematical problems. For example, solitons or other non-perturbative effects in QFT cannot be seen at any ...
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240 views

In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...
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84 views

A question on page 65 of Weinberg's QFT volume 1

The equation (2.5.12) on page 65 says that: $$ \left(\boldsymbol{\Psi}_{k',\sigma'},\boldsymbol{\Psi}_{k,\sigma}\right)=\delta^3\left(\boldsymbol{k}'-\boldsymbol{k}\right)\delta_{\sigma '\sigma}. $$ I ...
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1answer
94 views

Fermi Energy Variation

What would be a good Internet link that would properly explain Fermi Energy? How does the Fermi Energy of a material vary with external influence, such as doping of the material, and applied ...
4
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192 views

Spin-statistics theorem proof details

Recently I have read one book where there was some incomprehensible proof of the Pauli's spin-statistics theorem. I want to ask about a few details of the proof. First, the author derives ...
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42 views

Is Romer's letter on our search for the elementary proof of the spin-statics theorem out of date today?

The following link provides a letter to the editor by Robert H. Romer who writes, In a 1994 "question" in this journal, Neuenschwander asked whether anyone had yet met Feynman’s challenge of ...
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35 views

Sum of Green's functions in Condensed Matter

I am working on the Ginzburg-Landau model for Charge density waves, and I am carrying out the sum of Green's functions to calculate the terms in the GL model. I have the following question: Is the ...
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1answer
60 views

Interacting Lagrangian - Coupling constant and cutoff factor

I have a general question concerning a given interacting Lagrangian: $$\mathfrak{L}_I = \frac{g}{\Lambda^2} \bar{\chi} \ \gamma^\mu \gamma_5 \ \chi \ \partial^\nu F_{\mu\nu}$$ where $F_{\mu\nu}$ is ...
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111 views

Derivation of the full generator of the lorentz transformations

Let us study the subgroup of the Poincare group that leaves the point $x=0$ invariant, that is the Lorentz group. The action of an infinitesimal Lorentz transformation on a field $\Phi(0)$ is $L_{\mu ...
4
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1answer
147 views

Noether's Theorem: Foundations

I'm wondering on what principles Noether's theorem foots. More precisely: The action is a functional on the fields only. Why do we consider then variations of the space time too? In principle careful ...
2
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1answer
116 views

Do particles travel backward in time in a particle interpretation of field theory?

In this Phys.SE answer Ron Maimon stats: there is no relativistic particle formalism in which the particles have postive energies and casual propagation. You can either deal with fields in which ...
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65 views

Constructing Ward identity associated with conserved currents

Consider constructing the Ward identity associated with Lorentz invariance. It is possible to find a 3rd rank tensor $B^{\rho \mu \nu}$ antisymmetric in the first two indices, then the stress-energy ...
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436 views

How does Annihilation work?

How does annihilation work? I'm wondering why matter and antimatter actually annihilates if they come into contact. What exactly happens? Is that a known process? Is it just because of their different ...
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1answer
366 views

Physical reason for annihilation? [duplicate]

What is the fundamental reason as to why matter and antimatter annihilate? Is it because both particles and antiparticles are excitations of quantum fields, and the annihilation process corresponds ...
2
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1answer
59 views

Quantum numbers in QFT

In nonrelativistic quantum mechanics the state of a system is characterized by a vector of a Hilbert space. To characterize a state we need a complete set of commuting observables, and once we have ...
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1answer
134 views

What is the meaning of the negative vacuum expectation value of the Higgs field? Do we see it in nature?

In studying about the Higgs field and related, I find little mention of the equilibrium point at -V. I would like help conceptualizing what a negative vacuum expectation value is, ideally with respect ...
3
votes
1answer
154 views

Time-orded operator in Srednicki

On page 51 Srednicki states, "Note that the operators are in time order...we can insert $T$ without changing anything". This I agree with. But then on the next paragraph he states "The time order ...
4
votes
2answers
92 views

Poincare Generators in terms of Position and Momentum

The $10$ generators of the Poincare group $P(1;3)$ are $M^{\mu\nu}$ and $P^\mu$. These generators can be determined explicitly in the matrix form. However, I have found that $M^{\mu\nu}$ and $P^\mu$ ...
5
votes
3answers
376 views

How does QFT help with entanglement?

I'm a bit confused. QFT is claimed to incorporate both Quantum Mechanics and Special Relativity. Therefore it should address the problem of non-locality caused by entanglement. However when I search ...
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32 views

Stability Group of the Poincare Group

The stability group $G_\Sigma$ is a subgroup of the Poincare group $P(1;3)$. Its generators $X$ in the front form leave the hypersurface $\Sigma: x^+ = 0$ invariant. Phrased differently they satisfy ...
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2answers
183 views

How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
10
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1answer
194 views

Polar Decomposition of a Complex Scalar Field

People often write a complex scalar field via polar decomposition. What does this parametrization precisely mean? To be more explicit consider the following Lagrangian of a complex scalar field with ...
0
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1answer
77 views

Fourier expansion of the Klein-Gordon field

Is there a reason(both physical and mathematical) why the Klein-Gordon field is represented as a fourier expansion in the second quantization as opposed to other mathematical expansions? Be gentle ...
4
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1answer
208 views

Determinant for a coupled fluctuation Lagrangian

Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
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54 views

why the pole of scalar current correlator imply a dilaton?

In one recent paper, the author says the massless pole of fermion-antifermion scalar current correlator $< {J_0}{J_0} >$ implies a dilaton, where ${J_0} = {\psi ^ + }\psi (x)$. Above (3.18) he ...
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votes
1answer
79 views

Correlation functions and connection to ward identities

I have the following definition of a general correlation function $$ \langle \Phi(x_1)\dots \Phi(x_n)\rangle = \frac{1}{Z} \int [d\Phi] \Phi(x_1)\dots\Phi(x_n)e^{-S[\Phi]} $$ I have only just ...
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1answer
61 views

Operator Product Expansion in Massless 2D QED

In Peskin & Schroeder chapter 19 page 656, where the axial current anomaly of massless 2D QED is discussed, the authors go from: $$ ...