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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227 views

Why do we require quantum fields to vanish at infinity?

Classical fields, like the electrical field must vanish at infinity, because otherwise their energy would be infinite. This can be used in computations to exclude certain solutions. In quantum ...
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113 views

Renormalization condition

Can any on explain to me, why renormalization condition $$\Sigma(\gamma_\mu p^\mu=m)=0,$$ for one loop implies $$\Sigma_2(m)=m\delta_2-\delta_m~?$$ In the original $\Sigma_2$ we had $ m_0$ which is ...
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1answer
364 views

How do I solve this Gaussian path integral?

Suppose $$ Z = \int \mathcal D[\phi^*] \mathcal D[\phi] \exp(\phi^*A\phi + \phi B\phi) $$ where $A$ and $B$ are operators. I know how to solve a Gaussian path integral involving only $\phi^* A \phi$ ...
2
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1answer
67 views

In what sense is the chiral decomposition of spinors unique?

We may decompose a spinor field $\psi = \psi_L + \psi_R$ where $\psi_L = \frac12 (1 - \gamma^5) \psi$ and $\psi_R = \frac12 (1 + \gamma^5) \psi$. (I believe this is because the clifford algebra has ...
8
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1answer
215 views

What is the quantum state of a static electric field?

This is something that I've been curious about for some time. A coherent, monochromatic electromagnetic wave is well described by a coherent state $|\alpha\rangle$. The quantum treatment of the ...
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81 views

Matter antimatter fundamental and adjoint representation (Hermitian Anti-Hermitian)

I’m struggling with the following. I read in “The Standard Model: A Primer by Cliff Burgess”, page 493, that fermion fields in the fundamental representation can be thought of as column vector(s) ...
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1answer
32 views

Chrial multiplet's fundamental and anti-fundamental representation

Here i follow the notation in arXiv 9312104v1 (Witten's Verlinder algebra ~ paper) The usual kinetic energy for a chiral multiplet is given as (In 2 dimensional $N=(2,2)$ supersymmetry theory) ...
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85 views

Field renormalization of scalar Yang-Mills

In most books, one can find the field renormalization $Z_3$ in Yang-Mills with fermionic matter in the fundamental. In the $\overline{MS}$ scheme, tt is given by $$ Z_3 = 1 + \frac{g^2}{16\pi^2 ...
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70 views

How to deal with coupled fermion boson operators?

I am a beginner in field theory and I have an exercise where I have a product of coupled fermion boson operators? $$ \hat{b_{l} }^{\dagger}\hat{c_{l^{'}} }^{\dagger}\hat{a_{q} }\hat{b_{l} ...
4
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1answer
111 views

Number of Goldstone bosons in paramagnetic-to-ferromagnetic phase transitions

In paramagnetic-to-ferromagnetic phase transitions, the symmetry spontaneously breaks down from SO(3) to the subgroup SO(2) below $T_\text{crit}$. This implies that there should be two Goldstone modes ...
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222 views

What exactly do we mean by symmetry in physics?

I'm referring here to invariance of the Lagrangian under Lorentz transformations. There are two possibilities: Physics does not depend on the way we describe it (passive symmetry). We can choose ...
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1answer
126 views

One loop tadpole diagram $\phi \to \phi$ in $g\phi^3$ theory

I am trying to evaluate the tadpole diagram of $\phi^3$ theory to practice one loop amplitudes, but I am stuck at a certain point. The amplitude is given by the integral, $$\mathcal{M} = ...
3
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2answers
136 views

Global symmetry and particle multiplets

In chapter 20, of Peskin and Schroeder's quantum field theory book, they start with a comment that a global symmetry that is manifest lead to particle multiplets with restricted interactions. Can ...
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23 views

Why third Pauli $\tau_3$ becomes third Isospin component $\tau_3^{<\Phi>}$?

When considering the higgs coupling to the neutral gauge boson of EW theory (see e.g. C. G. Tully (EPP nutshell) page 102): $$\tag{1}\mathcal{L} = \frac{1}{4}\left\{\left(g' B_\mu Y_\Phi+gW_\mu^3 ...
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1answer
322 views

How does QFT interpret the Negative probability problem of the real scalar fields' Klein-Gordon equation?

I am totally a beginner in QFT, here's the problem that I got: for the real scalar fields, are there any elementary particles descriped by them. If so, how to understand the negative probability ...
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44 views

Why scattering of red and blue quark only involves $G_8^\mu$?

According to the author C. G. Tully (Particle physics in a nutshell), the scattering of a red and blue quark only involves $G_8^\mu$. How come this is so? I thought $G_3^\mu$ and $G_8$ only mediate ...
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1answer
95 views

What would be the most general effective Lagrangian involving one Higgs and two gluons?

Two different possibilities come into my mind $\mathcal{L}\sim{}HG_{\mu}G^{\mu}$ where $G^{\mu}$ is the gluon field and $H$ the Higgs, or either $\mathcal{L}\sim{}HG_{\mu\nu}G^{\mu\nu}$ Where ...
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55 views

Evaluation Feynman parameters from denominator

I try to evaluate Feynman parameters but got stuck at some point. $$ \int_0^1 \frac{1}{(Ax+(1-x)B)^2}\,dx=\frac{-1}{(Ax+B(1-x))}\frac{1}{A-B}=\frac{1}{AB} $$ $$ \frac{1}{AB}=\int_0^1 \int_0^1 ...
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0answers
68 views

't Hooft many instanton solutions

I'm study 't Hooft many instanton solutions of self-duality equation. In this method $A^a_\mu=-\bar{\eta}^{a}_{\mu\nu}\partial^\nu \ln{\Phi}$. After substitution in self-duality equation I've proven ...
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102 views

Klein-Gordon field commutator integral identity [closed]

Consider a Klein-Gordon field $\phi$ on points $x,y$ of $\mathbb R^4$ Minkowski-spacetime. Here I'm writing $x=(x^0, \stackrel \rightarrow x)$ so that $\stackrel \rightarrow x$ gives the spatial ...
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3answers
179 views

Klein Gordon for spin-1 particle photon

If Klein Gordon equation is for spin-0 particles, I write massless fields as $\square A=0$, how can I say $A_\mu=\epsilon^\mu e^{-ikx}$ as a wave function of polarized photon (spin-1) ?
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1answer
1k views

Difference between 1PI effective action and Wilsonian effective action?

What is the simplest ay to describe the difference between these two concepts, that often go by the same name?
3
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1answer
74 views

Variation of the kinetic quark term of the QCD Lagrangian under gauge transformation

A simple kinetic quark term would look like $$\bar{\psi}(\gamma^{\mu}\partial_{\mu} - m){\psi}.$$ Imposing SU(3) symmetry the Dirac spinor transforms like $$\psi(x) \rightarrow \psi'(x) = e^{ig_s ...
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105 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, ...
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2answers
247 views

Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...
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3answers
3k views

Noether theorem, gauge symmetry and conservation of charge

I'm trying to understand Noether's theorem, and it's application to gauge symmetry. Below what I've done so far. First, the global gauge symmetry. I'm starting with the Lagragian ...
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2answers
84 views

SU(2) kinetic term as a trace

Is there a easy way to rewrite the SU(2) kinetic term as a trace? As in $$\mathcal{L} = -\frac{1}{4}\vec{F}_{\mu\nu}\vec{F}^{\mu\nu}\\[1cm] = -\frac{1}{2}\mathrm{tr}\Bigg[\bigg(\vec{F}_{\mu\nu}\cdot ...
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0answers
32 views

A question about the interchanges of particles belonging to species in Weinberg's QFT book 1

Weinberg put this in page 171 that I can't quite understand: If we like, we can avoid this question by simply agreeing from the beginning to label the state-vector by listing all photon momenta and ...
5
votes
1answer
123 views

Is there a way to get the spin naturally in nonrelativistic theories?

We all know how spin is added in a rather ad-hoc way in quantum mechanics. In the other hand, in relativistic quantum field theories the spin structure arises quite naturally from the fields. Is ...
2
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2answers
104 views

Group representation acting on operators (QFT)

I have found in many texts the following statement: Let $T_g$ be a representation of a group (of transformations, e.g. rotations, translations, Lorentz transformations ) acting on a given Hilbert ...
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2answers
69 views

Showing that a bilinear variation is Lorentz invariant

Let $\psi, \chi$ be a spinor (say Dirac). Then the infinitesimal Lorentz variation is given by $$\delta \psi = -\frac{1}{4}\lambda^{\mu \nu} \gamma_{\mu \nu}\psi$$ then I think that the conjugate is ...
8
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1answer
165 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 ...
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0answers
71 views

Physical significance of Dirac equation in (2+1)-D

What's is the physical significance of the two inequivalently irreducible-represented Dirac equations in (2+1)-D? As it is known, all the $4\times 4$ matrix representations of the Dirac algebra ...
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37 views

Variational calculus needed for QFT [duplicate]

Where can one learn the variational calculus needed for QFT? Im not sure a whole book of super rigorous treatment is what i need.
1
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1answer
68 views

Baryogenesis via Leptogenesis

Baryon number is directly violated through electroweak anomaly and so does the Lepton number, for each transition from one vacuum to another. The two violations are of equal amount $\Delta B=\Delta ...
3
votes
2answers
486 views

What's the boundary of microscopic world and macroscopic world?

In other words, what's the maximum size of a Quantum denizen upto which it shows Quantum behaviors? How big do I need to create a molecule (or, collection of molecules) so that Feynman's multiple path ...
3
votes
1answer
122 views

In QFT, do the fields evolve with determinism, in principle?

In quantum mechanics, the outcomes of a certain measurement might not be deterministic. However, the wavefunction evolves with determinism according to Schrodinger's equation. Is QFT analogous in ...
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2answers
271 views

Why would electrons have Weak Charge? [closed]

Electrons (and, their cousins Muon and Tau) carry Weak Charge having value $-1/2$. If you believe in Strong Anthrophic Principle Why does electrons carry Weak Charge? If you don't believe in Strong ...
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0answers
66 views

Colour decomposition in QCD

I am looking to compute the matrix element for the process gg -> u ubar at leading order. It is straightforward to calculate the non-colour part of the usual s, t and u channels. I will call these ...
2
votes
1answer
91 views

Quick non rogorous way to obtain Feynman rules from a Lagrangian in a non abelian theory

I have been told that a quick way to get the Feynman rules from a Lagrangian is to take an interaction term, forget about the fields and multiply an $i$. This works perfectly for example for QED but I ...
3
votes
1answer
236 views

Why is there sometimes an additional term in the orthogonality relations for the polarization vectors?

When considering the polarization vectors of a massive spin-1 field, like an $A_\mu$ with Lagrangian density $$ \tag{A} \mathscr{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{2}M^2 A_\mu A^\mu,$$ ...
0
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1answer
56 views

Relation between $a_{-p}$ and $a^{\dagger}$? for real scalar field

Fast question. Consider the real Klein-Gordon field. Is there a way to relate $a_{-p}$ with $a_p^{\dagger}$?
2
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0answers
81 views

Stack presentations and massive non-conformal theories

In the paper, Cluster Decomposition, T-duality, and Gerby CFT’s , by Hellerman, Henriques, Pantev and Sharpe, in the introduction it says: "Briefly, the idea is that nearly every stack has a ...
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1answer
165 views

Renormalization in Classical Field Theory

1) The statement that general relativity (GR) is not renormalizable - is it a statement only about the quantization of GR or is it non-renormalizable also as a classical field theory? 2) More ...
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3answers
270 views

Why gauge $SU(N)$ and not $SO(N)$?

When building models people typically gauge $SU(N)$ but rarely try to gauge $SO(N)$ (the only example I know about is $SO(10)$, but even that isn't quite $SO(10)$ but actually its double cover). At ...
4
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2answers
278 views

Where does the electric force come from if an electron has no definite location?

Say electron A is nearby another electron (B), so that they may repel each other. Electron B is in a position eigenstate (so it has a definite position). But electron A is not. How does electron A ...
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1answer
218 views

Why does renormalization need an unbroken symmetry?

Common wisdom is that for a QFT to be renormalizable it must be invariant under a symmetry transformation. Why does renormalization need an unbroken symmetry? Which is the first publication that ...
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1answer
94 views

Phase space measure in positron-electron annihilation calculation

I'm still trying to calculate the cross-section of the $e^- e^+ \rightarrow \mu^- \mu^+$ interaction in first order. This time I'm struggling with the phase space measure. Note that I have two ...
9
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1answer
945 views

Why should bosons be in the adjoint representation of the gauge group?

Is there a deep mathematical reason for why bosons should be in the adjoint representation of the gauge group rather than any other representation?
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59 views

Conservation of spin angular momentum at the vertex having two fermions and graviton

Consider an electron positron scattering/annihilation. This process can also take place via gravitons (as a propagator) because fermions have mass. But the graviton has spin angular momentum of 2 ...