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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Is a lagrangian with a background field interaction renormalizable ? If yes, when?

Consider the Lagrangian, $$ L = -\partial_{\mu} \chi \partial_{\nu} \chi^{\dagger} - m^2 \chi \chi^{\dagger} + g\chi \chi^{\dagger}\phi,$$ where $\phi$ is a background field and $\chi$ is a complex ...
6
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1answer
324 views

Casimir forces and its associated Feynman propagator

This is a continuation to my previous question, in which I began an attempt solve the Casimir Force problem using path integrals. As one of the answers there suggest I solve the Feynman propagator ...
33
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3answers
3k views

Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
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0answers
80 views

Transformation properties of $2 \times 2$ matrix involving Pauli matrices

Suppose the vector $\phi$ transforms under $SU(2)$ as: $$\phi' = (\exp(-i \alpha \cdot t))_{ij}\phi_j,$$ where $(t_j)_{kl} = −i \epsilon_{jkl}$ and $j, k, l \in \left\{1, 2, 3\right\}.$ Based on ...
7
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2answers
109 views

Lie groups with same algebra

I had a problem when considering symmetry breaking in an SO(4) gauge theory: $\mathcal{L} = \left| D_\mu\phi \right|^2$ where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some ...
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1answer
94 views

Does massive particle really move at speed of light? [closed]

According to this answer I understood that particles with mass also move at speed of light but interaction with higgs field make this movement zigzag. So average speed is below speed of light. But I ...
2
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0answers
32 views

Contour integral of the retarded Klein Gordon propagator

I've been trying to prove by hand the Peskin's formula for the retarded propagator of the Klein Gordon equation, that is, $$\int_{x^0 > y^0} \frac{d^4p}{(2\pi)^4} \frac{-e^{-ip(x-y)}}{i(p^2 - ...
3
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1answer
72 views

Quantum field theory: zero vs. finite temperature

I have recently been made aware of the concept of thermal field theory, in which the introductory statement for its motivation is that "ordinary" quantum field theory (QFT) is formulated at zero ...
3
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0answers
32 views

Has anyone studied anomalous supersymmetry?

In this paper (and others), the authors study a supersymmetric model where the supercharge suffers an ABJ anomaly. Has anyone studied a supersymmetry with a 't Hooft anomaly (gauging the ...
3
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2answers
73 views

Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
3
votes
2answers
231 views

Do Standard Model Yukawa couplings depend on the gauge choice?

In the standard model and the Unitary gauge, we write the Higgs field as $ \phi = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v + H \end{pmatrix}$ and the Yukawa couplings (leaving out the neutrino ...
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1answer
60 views

Cutoff-dependent “inverse propagator” for renormalization

In Zee's QFT in a Nutshell, when introducing mass renormalization, he calculates the "inverse propagator" for a $\phi^4$ scalar field theory to order $\lambda^2$ by considering the two diagrams shown: ...
3
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1answer
538 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 ...
3
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1answer
57 views

Is Lorentz invariant differential measure arbitrary?

In Srednicki, we chose a function $f(\mathbf k)$ to make $d^3\mathbf k/f(\mathbf k)$ Lorentz invariant. The way to do this is to first start from a 4 dimensional measure and multiply it by a Dirac ...
14
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0answers
241 views

How to apply the Faddeev-Popov method to a simple integral

Some time ago I was reviewing my knowledge on QFT and I came across the question of Faddeev-Popov ghosts. At the time I was studying thеse matters, I used the book of Faddeev and Slavnov, but the ...
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43 views

Gravity modeled by warping of spacetime or by field field theory?

I've recently read "Fields of Color" by Rodney Brooks who states that there are currently two ways of understanding the phenomenon of gravity. One involves a warping of 4D spacetime a la Einstein, ...
6
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1answer
274 views

Counterterm Lagrangian and Renormalisation?

I am going through the notes on QFT by M. Srednicki (online: http://web.physics.ucsb.edu/~mark/qft.html), and I am having a hard time to understand the "renormalised" Lagrangian. Consider a ...
7
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0answers
53 views

Why would renormalization be necessary without divergent integrals? [duplicate]

Weinberg uses the LSZ reduction formula to introduce field renormalization,and on page 441, he says: As this discussion should make clear: the renormalization of masses and fields has nothing to ...
5
votes
2answers
121 views

Is the Noether charge always a Hermitian operator?

Noether's theorem tells us that to every continuous symmetry of the Lagrangian there corresponds a conserved current $j^\mu$. From the time component of this current, we can then define the Noetherian ...
0
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0answers
25 views

Electroweak instanton calculations

Consider an electroweak instanton in a model beyond the Standard Model with explicit baryon plus lepton number ($B+L$) violation. This instanton decays into nine quarks $q$ and three leptons $l$, ...
3
votes
0answers
25 views

Polarization vectors in Quantum Electric Field

The quantum electric field is written as, \begin{equation} \mathbf{E}(\mathbf{r})=i\sum_{\mathbf{k},\lambda}\sqrt{\frac{\hbar \omega}{2 V ...
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4answers
237 views

Is there some special case where a fermion can mediate a force?

Looking at the comments of this questions Does the gravitino contribute to the gravitational interaction? and even considering that the answers here in this other question Why are all force particles ...
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1answer
99 views

A question on using Fourier decomposition to solve the Klein Gordon equation

Given the Klein Gordon equation $$\left(\Box +m^{2}\right)\phi(t,\mathbf{x})=0$$ it is possible to find a solution $\phi(t,\mathbf{x})$ by carrying out a Fourier decomposition of the scalar field ...
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0answers
23 views

relation between operator and matrix

Recall that in quantum mechanics, the three components of s of a spin-$\frac{1}{2}$ particle satisfied the anticommute relation: $$ \{s^i, s^j\}=\delta^{ij} $$ and we could parametrize the operators ...
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1answer
179 views

One-Loop Yukawa RGEs

I'm currently trying to understand how one can write the one-loop RGEs for the Yukawa couplings using the general formula: One example I'm interested in is how the author derives, using this ...
3
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1answer
74 views

Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially?

I read several textbooks of QFT and find that there are two ways to classify the particles or fields. The first one is to study the irreducible representation of Lorentz group (or exactly the ...
2
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1answer
93 views

S-matrix element

I'm confused with the relation between the fully resummed propagator in a given QFT and the corresponding S-matrix element. According to the LSZ reduction formula ($\phi^4$ theory for definiteness ...
3
votes
1answer
46 views

Feynman Propagator in Peskin & Schroeder

To prove Wick's Theorem, Peskin & Schroeder define the contraction of two fields: \begin{align} \text{Contract}[\phi(x)\phi(y)]\equiv \begin{cases} [\phi^+(x),\phi^-(y)] & \text{for ...
2
votes
1answer
62 views

What is the axial current?

The axial current is defined as $$j^\mu_5 = \bar{\psi} \gamma^\mu \gamma_5 \psi.$$ This quantity is important when studying anomalies. Explicitly working out components, the axial current is just the ...
5
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3answers
2k views

Dirac spinor and Weyl spinor

How can it be shown that the Dirac spinor is the direct sum of a right handed Weyl spinor and a left handed Weyl spinor? EDIT:- Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed ...
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7answers
11k views

What is a complete book for quantum field theory? [closed]

I am searching for a complete and comprehensive book for QFT. What is, in your opinion, a good one?
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0answers
55 views

Decay width in 2 & 3 body decays, calculating momentum integrals

I'm considering a toy model with two types of scalar particles, one massive $(\Phi)$ and one massless $(\phi)$ with an interaction of the form $$L_{int}=-\lambda \phi\phi\Phi$$ I'm interested in a ...
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0answers
28 views

Single particle state in $\phi^4$ theory

I'm quite happy with the idea that a multi-particle state in a free scalar field theory has a discrete energy spectrum, and that turning on a quartic coupling $\frac{\lambda}{4!}\phi^4$ acts as a ...
5
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1answer
289 views

Can the Higgs condensate be described in terms of creation operators?

In superconductivity, the BCS condensate can be described in terms of 2 creation operators (the 2 electrons of the pair) acting on the vacuum. I'm wondering whether a similar description can be given ...
2
votes
1answer
86 views

Quantization of free real scalar massless field in 2d

Is there a reference to literature where one explicitly constructs quantization of the free real scalar massless field in the 2-dimensional space-time? In particular, how the propagator looks like? ...
1
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1answer
123 views

Invariance in Euclidean and Minkowski spaces

Consider Wick's rotation from Minkowski to Euclidean space in QFT. What is the connection between O(4) invariance in Euclidean space and Lorentz invariance in Minkowski space? If we define a quantity ...
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1answer
67 views

Replica trick for calculating Entanglement Entropy?

This is probably a simple question. Von Neumann entropy is defined to be $$S_A=-tr_A\rho_A \log\rho_A$$. And it's said that it can be calculate from the "Replica trick": $$S_A=\lim_{n\to 1}\frac{tr_A ...
7
votes
2answers
752 views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. ...
2
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0answers
65 views

Feynman rules from interaction Lagrangian with electromagnetic tensor (vertex)

I am currently studying for my QFT exam and in particular learning the methods of reading the Feynman rules directly off the Lagrangian. However, I'm still a bit uncertain how to deal with ...
3
votes
1answer
847 views

What is a quantum field?

I'm just tasting a bit of QFT and want to get started. I got stuck right at the start: what is a quantum field, and how should I look at it? This question can be a follow up of the What is a field, ...
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0answers
34 views

Confusion over trying to understand spinor components

I've been reading about the quantisation of the Dirac field $\psi(x)$ and it is stated that the general solution to the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi(x)=0$ is given by the ...
1
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1answer
70 views

Tetrad choice for Pauli-Lubanski in the massless case

The Pauli-Lubanski pseudovector coincides with intrinsic spin in the rest frame of the particle. In a more general frame, one defines a tetrad and projects the PL vector on it to define intrinsic spin ...
3
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1answer
280 views

effective field theory of the projective semion model

The "projective semion" model was considered in http://arxiv.org/abs/1403.6491 (page 2). It is a symmetry enriched topological (SET) phase. There is one non-trivial anyon, a semion $s$ which induces a ...
2
votes
1answer
43 views

Why is the symmetric phase in a Bose gas not superfluid?

In the theory of superfluidity in weakly interacting Bose gases, one finds that in the symmetric phase the exctitations have the dispersion relation $\omega = \frac{k^2}{2m}-\mu$ with gap ...
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0answers
28 views

can different force fields interfere (create interference patterns)

Edit: I have rewritten the question for clarity. I know waves of photons can interfere eachother. What about if you mixed waves of photos with w and z bosons? What about gravitons (if they exist) ...
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0answers
32 views

Obtaining the $s,t,u$ Feynman diagrams by Wick contraction

Consider a real scalar field described through the following lagrangian $$\mathcal L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}m^2 \phi^2 - \frac{g}{3!}\phi^3$$ The second ...
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0answers
30 views

$\mathcal{N} = 4$ Super-Yang Mills propagators

In $\mathcal{N} = 4$ Super-Yang mills there are only massless particles. If one wishes to obtain a heavy quark one can see the SYM theory as a stack of (N+1)-branes in AdS$_5 \times$S$^5$ where one ...
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0answers
33 views

Why is the bosonic Casimir force attractive while the fermionic CF is repulsive?

Why is the bosonic Casimir force (CF) attractive while the fermionic CF is repulsive? I looked up for many papers and books but all are focused on boundary conditions. I didn't find any conceptual ...
3
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0answers
60 views

How to find propagator from Lagrangian at a glance?

If I have a Lagrangian in momentum space of the form $$ \mathcal{L} = W_\mu^{ \dagger}(p)f(p)^{\mu \nu}W_\nu(p) $$ how is the propagator for the field related to the function $f(p)$ (e.g. is it ...