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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How (why!?) does one introduce an UV cut-off in dimensional regularization?

This question is in reference to the confusing equation 3.7 (page 14) of this paper. One sees the 1-loop answers in their theory as given in their A.7 and A.8 on page 20. Each of the terms is a ...
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233 views

Quantum master equation in the Batalin-Vilkovisky formalism

I am reading the Section 15.9 of Weinberg's book "The Quantum Theory of Fields, vol. 2". Under a shift $\delta\Psi[\chi]$ in $\Psi[\chi]$, we have $$ \begin{split} \delta Z&=i\int[d\chi]\...
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Why do single particle states furnish a rep. of the inhomogeneous Lorentz group?

Following up on this question: Weinberg says In general, it may be possible by using suitable linear combinations of the $\psi_{p,\sigma}$ to choose the $\sigma$ labels in such a way that $C_{\...
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286 views

Physical Interpretation of Lorentz-transformed Single Particle states being linear

As in this question, let $\psi_{p,\sigma}$ be a single-particle 4-momentum eigenstate, with $\sigma$ being a discrete label of other degrees of freedom. Weinberg discusses the effect of a homogenous ...
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79 views

Why are non-momentum DoFs of single-particle states discretely labeled?

Following the treatment of Weinberg, chapter 2, we consider $\psi_{p,\sigma}$ as single-particle eigenstates of the 4-momentum. Weinberg says that $\sigma$ labels all other degrees of freedom and we ...
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What is energy in $z \neq 1 $ theories?

In a critical theory with dynamical critical exponent $z \neq 1 $, which amongst frequency, $\omega$, and dispersion, $E(\vec{k})$, may be referred to as ''energy''? I'm confused about this since in ...
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832 views

What goes wrong when one tries to quantize a scalar field with Fermi statistics?

At the end of section 9 on page 49 of Dirac's 1966 "Lectures on Quantum Field Theory" he says that if we quantize a real scalar field according to Fermi statistics [i.e., if we impose Canonical ...
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275 views

Anti-particle problem for Dirac sea

According to the Dirac hole theory we know that Dirac sea is completely filled with negative energy, called vacuum. We will need $2mc^2$ or greater to get electron and a positron by incident photon. ...
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2k views

Unitary quantum field theory

What do physicists mean when they refer to a quantum field theory being unitary? Does this mean that all the symmetry groups of the theory act via unitary representations? I would appreciate if one ...
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582 views

What is the reason why anyons escape spin-statistic theorem?

I'm wondering about the exact reason why anyons escape the spin-statistic theorem (SST), see e.g. http://en.wikipedia.org/wiki/Spin–statistics_theorem. I've read somewhere (the wikipedia page is ...
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175 views

Alternative interpretation of Off-shell internal QFT propagators?

In Quantum Field Theory in a (1, D - 1) space-time, to calculate transition amplitudes, we are using Feynman diagrams, where internal lines (internal propagators) corresponds to momenta which are said ...
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295 views

Complex masses for Dirac and Weyl spinors

I'm trying understand how to rotate Dirac fields to absorb complex phases in masses. I have a few related questions: With Weyl spinors, I understand, $$ \mathcal{L} = \text{kinetic} + |M|e^{i\...
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Gauge fields and strings: Loop equations

I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \delta \Psi (C) ~=~ \int_{0}^{2\pi} {\rm P} \left(F_{\mu\nu}(x(s)) \exp \oint_C A_\mu dx^\mu \right)\dot{x}_\nu \delta x_\mu(x) \, {\...
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70 views

Relevant operators in two dimensional O(n) models

The most general hamiltonian of a two dimensional $O(n)$ and $Z_2$ invariant statistical model can be written: $$ H=\int d^2 x \left[\frac{\nabla \mathbf{\phi}^2}{2} + \frac{m_0^2}{2}\mathbf{\phi}^2 +\...
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293 views

Quantum field theory quote

I have read this in scientific American: According to quantum field theory, all particles spend a little time as combinations of all other particles" Is this right? How long? And how can they be a ...
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578 views

Quantum Field Theory and Hilbert space dimensionality

Much (All?) of quantum theory can be done in separable Hilbert spaces with a countable basis. How about quantum field theory? Is it “quite happy” (mathematically consistent) if everything is ...
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831 views

Can one define a “particle” as space-localized object in quantum field theory?

In Peskin and Schroeder, while discussing creation and annihilation operators for a Klein-Gordon field (p.22), the authors say, as we all know the creation operator $a_p^{\dagger}$ acts on vacuum to ...
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765 views

How to directly calculate the infinitesimal generator of SU(2)

We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial U}{\...
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1answer
172 views

Lie algebra of lorentz group

I'm stuck in following calcualtion from sredniki's QFT book.(Its actually in the solution manual) How can i get from $$\delta\omega_{\rho\sigma}(g^{\sigma\mu}M^{\rho\nu} - g^{\rho\nu}M^{\mu\sigma}) $...
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Categorizing solutions to Hierarchy problem

We know that no gauge symmetry can prevent a term $m_\phi^2|\phi|^2$ for a scalar field, and that, given the quadratic loop corrections, the natural scale is $m_\phi \sim M_P$. This is related to the ...
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246 views

What is the reason that relativistic corrections for hydrogen atom work?

Here I cite part from Sidney Coleman's lectures on Quantum Field Theory: It is a phenomenal fluke that relativistic kinematic corrections for the Hydrogen atom work. If the Dirac equation is used, ...
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582 views

Gradient involved commutator in $\phi^4$ theory

In a phi fourth theory, the Hamiltonian density is: $$\mathcal{H}=\frac{1}{2}\pi^2+\frac{1}{2}(\nabla \phi)^2+\frac{1}{2}m^2\phi^2+\frac{\lambda}{4!}\phi^4$$ Now I impose the usual equal time ...
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Calculating the the kernel using path integrals for quadratic lagrangians

I am reading Feynman and Hibbs on Path Integrals. In section 3.5, they show that the kernel for a lagrangian of the form $L=a(t)\dot{x}^2+b(t)\dot{x}x+c(t)x^2+d(t)\dot{x}+e(t)x+f(t)$ is $K(b,a)=e^{\...
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367 views

Imaginary time in QFT

I'm reading chapter 4 of Introduction to Quantum Field Theory by Peskin & Schroeder. In the $\phi^4$ theory, the authors state that the ground state of the interaction theory $|\Omega\rangle$ can ...
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Scalar-fermion bound state

Is it possible to have a bound state between a scalar and a fermion? For example, a squark--anti-squark bound state, provided that the decay width is sufficiently small compared to the binding energy?
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Derivation of Dirac equation using the Lagrangian density for Dirac field

How can I derive the Dirac equation from the Lagrangian density for the Dirac field?
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306 views

Volume element $\mathrm{d}^4k =\mathrm{d}k^0 \,|\mathbf{k}|^2\,\mathrm{d}|\mathbf{k}| \,\mathrm{d}(\cos\theta) \,\mathrm{d}\phi$ in Minkowski space?

Suppose we have an integral $$\int \mathrm{d}^4k \,\ f(k)$$ we want to evaluate and that we're in Minkowski space with some metric $(+,-,-,-)$. Is it true that: $$\mathrm{d}^4k = \mathrm{d}k^0\ \...
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362 views

de Sitter versus Minkowski QFT and cosmological constant

WMAP/Planck results confirm than we live in a de Sitter-like phase, i.e., a Universe with positive acceleration or positive cosmological constant! Therefore, I believe that a way to solve the ...
7
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2answers
399 views

Mass gap for photons

I am puzzled by the answers to the question: What is a mass gap? There, Ron Maimon's answer gives a clear-cut definition, which I suppose applies to any quantum field theory with Hamiltonian $H$, ...
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109 views

Intuition behind the notion of reflection positivity

I came across Yuji's question. I'm finding it difficult to parse the meaning behind what's said on Wikipedia. Could someone give an explanation of the concept involved? I would also appreciate ...
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Bosonic-Fermionic interactions in supersymmetry

There are a lot of supersymmetric theories, and, sometimes,in the Lagrangian, there are interacting terms between bosonic and fermionic degrees of freedom, and sometimes not. Why ? For instance, for ...
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262 views

Beta-function non-zero at classical level?

In Jaume Gomis's lecture 5 on CFT at Perimeter Institute, he says (at 27:40 minute mark) that the beta function, classically, of the $m^2$ parameter in massive $\lambda \phi^4$ theory is $$\beta(m^2) ...
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226 views

Quantum field theory alternatives

Quantum field theory arises from the requirement that the S-matrix is lorentz scalar and obeys the cluster decomposition principle. I want to know if there are other ways to build invariant S-...
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98 views

Question regarding operators and cylindrical coordinates

I have the following problem in my hand: I need to arrive from the Cartesian expression $$x_{j}{\partial_{k}}x_{j}{\partial_{k}}-x_{j}{\partial_{k}}x_{k}{\partial_{j}}$$ to this expression: $$r^{2}\...
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Question on the Hagedorn tower in Type I string theory

In a previous question (Mass spectrum of Type I string theory), I had asked about the mass spectrum of Type I string theory. I got a response saying that it is a Hagedorn tower. However, my source ...
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1answer
382 views

Setting of renormalization scale in field theory calculations

In dimensional regularization an arbitrary mass parameter $\mu$ must be introduced in going to $4-\epsilon$ dimensions. I am trying to understand to what extent this parameter can be eliminated from ...
4
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1answer
146 views

Flavour diagonal SUSY breaking

Because there is a single Yukawa matrix for the SM leptons, the lepton mass and flavour states can be aligned, by diagonalization, even if the Yukawa matrix had off-diagonal elements. SUSY breaking, ...
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3answers
488 views

Many photons, one quantum field?

If a photon can be described as an excitation in a quantum field, is this the same field for all photons, or does each photon exist in its own field?
4
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152 views

Why doesn't one-photon-irreducible function have any pole at $q^2=0$?

I'm reading the QFT textbook by Weinberg. In volume one chapter 10 page 451, at the lower part of the page he says, Now, because $\Pi^*_{\mu\nu}(q)$ receives contributions only from one-photon-...
4
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326 views

For mesons, or baryons, do sea quarks contribute to the angular momentum of the bound state?

The total angular momentum of a bound state of quarks, such as a meson say, can be done by studying the spin and orbital angular momentum of the 2 valence quarks. What about the sea quarks why they ...
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About deriving the multi-trace index in terms of the single-trace index

This question is in reference to this paper Combining their equations 5.2, 5.3, 5.6 and 5.7 one seems to be looking at the integral/partition function, $Z(x) = \prod_{n=1}^{n =\infty}\left [ \int ...
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1answer
183 views

Parametrization of $U(N)$ non-linear sigma model

The motivation of this question actually comes from this (really old) paper of Weinberg. He considers a theory of massless pions. They have a chiral $SU(2)_{L} \times SU(2)_{R}$ symmetry. The pions ...
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2answers
306 views

How to prove that the generator of proper vertices is the Legendre transform of $W(j) = \log \frac{Z[j]}{Z[0]}$

I'm studying QFT from Le Bellac's book, but I can't understand very well his proof for the generator of proper vertices. Can someone give a more readable and/or understandable proof?
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Spectra of the Type II String theories

The spectrum of the Type II string theory (both IIA and IIB) is given by: \begin{array}{*{20}{c}} \hline & {{\text{Sector}}}& & {{\text{Spectrum}}}& & {{\text{Massless Fields}}} ...
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Four-gauge-boson vertex in non-Abelian gauge theories

In Peskin & Schroeder's book page 524, the following diagram is calculated for the gauge boson self-energy in order $g^2$: In dimensional regularization, its contribution is given by $$-g^...
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$\langle B|A \rangle$ expressed in terms of the Partition Function

Say you have an electron departing from point A and reaching poing B after a time t. According to some helping friend, the Partition Function for that electron going from point A to B can be written ...
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353 views

Can Divergences in Nonrenormalizable Theories Always Be Absorbed by (An Infinite Number of) Counterterms?

For example, consider the $\phi^3$ theory in $d=8$, with Lagrangian: $\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{1}{3!}\lambda_{3}\phi^{3}$. In 8 ...
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1answer
332 views

Energy eigeinstates written in the field operator eigenstates basis

For an harmonic oscillator we can write the Hamiltonian eigenvalues in the basis of the amplitude eigenvalues : for example the ground state is a gaussian : $⟨x|0⟩=a.e^{-b.x^{2}}$. I was wondering ...
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199 views

quantization of Dirac field

The general solution to the Dirac equation is a sum of plane wave solutions $$ \psi(x) \sim \int d^3k \sum_r b_r(k) u_r(k)e^{-ikx} + d^\dagger_r(k) v_r(k)e^{+ikx} $$ The basis spinors $u_r$ and $v_r$ ...
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Is conservation of statistics logically independent of spin?

If the number of fermions is $n$, we expect the quantity $(-1)^n$ to be conserved, i.e., $n$ never changes between even and odd. This is known as conservation of statistics. In the normal context of ...