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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S-operator lorentz invariance

How to show that $\hat {S}$-operator must be lorentz-invariant operator? $$ |\Psi (t)\rangle = \hat {S} | \Psi (0) \rangle , \quad \hat {S} = \hat {T}e^{-i\int \hat {H}_{I}d^{4}x}. $$ I have read ...
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1answer
39 views

Why is $\vert I=1,I_3=1\rangle = -p\bar n$

My book doesn't explain well how to build a doublet of antiparticles that transforms the same way the particle doublet $(p,n)^T$ (proton neutron) does. They claim $$\tag 1 \vert I=1,I_3=1\rangle = ...
3
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1answer
40 views

Effective theories and dimension six operators

What is the importance of dimension six operators in the study of physics beyond the Standard Model? Are these operators more relevant than dimension five operators like $HHFF$ or operators with ...
2
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1answer
39 views

What are threshold corrections?

As the title goes, what are threshold corrections in quantum field theory? In particular, I would be glad if a good reference is provided. Standard QFT books such as Peskin, Weinberg, etc seem to ...
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1answer
41 views

Branch cuts in two-point function

The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is ...
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38 views

Is $\overline{\psi_{L}^{c}}\psi_{R}^{c}=\overline{\psi_{R}}\psi_{L}$ true for two different spin 1/2 fermions?

In the context of seesaw mechanism or Dirac and Majorana mass terms, one often see the following identity $$ \overline{\psi_{L}^{c}}\psi_{R}^{c}=\overline{\psi_{R}}\psi_{L}. $$ Here, I am using 4 ...
2
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1answer
44 views

Goldstone couple to conserved current

The Goldstone in spontaneous symmetry breaking problem couples naturally to the associated conserved current of the broken symmetry. How can I see a rigorous (mathematical) derivation for that?
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2answers
68 views

Higgs mechanism and neutral fields

Consider a Lagrangian $L(\phi,A_{\mu})$ with $\phi$ being some scalar field and $A_{\mu}$ some dynamical U(1) gauge field that minimally couples to $\phi$. Under a global U(1) symmetry the field ...
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2answers
125 views

In what sense do Goldstone bosons live in the coset?

Goldstone's theorem says that if a group, $G$, is broken into its subgroup, $H$, then massless particles will appear. The number of massless particles are given by the dimension of the coset, $G/H$. ...
2
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1answer
48 views

Can weakness of gravity explore new dimensions

Since gravitational force is weakest force out of the four fundamental fources at the microscopic level. Is it possible that gravitational force is strong in a particular direction at a new ...
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26 views

Thermal propagator for a scalar field (KMS condition)

I'm having some troubles following the derivation of the scalar field thermal propagator. I'm following the article "Finite Temperature Quantum Field Theory in Minkwoski space" by Niemi and Semenoff ...
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25 views

Parity transformation of polarization vectors

In quantum field theory, for vector fields, we can write the following expansion: $$V^\mu = \sum_{p,\lambda} \left[ \epsilon^{\lambda, \mu}\left(p\right) a^\lambda\left(p\right)e^{-ip \cdot x} + ...
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63 views

QFT in curved space [on hold]

Can someone exactly tell me what one gains from doing QFT in curved space, and how reliable these new results are. I want to know if it is worth while putting some man hours towards this. Please ...
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39 views

Conflict between Lippmann–Schwinger equation and Gell-Mann and Low theorem about energy

Lippmann–Schwinger equation states that scattering state will have the same energy as free state, while Gell-Mann Low theorem says that they have different enery. Lippmann–Schwinger equation says: ...
3
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1answer
57 views

Photon Angular Momentum

Essentially I am wanting to evaluate $$\langle j\, m \mid a^\dagger(\mathbf{k}, \lambda) \mid 0 \rangle \,,$$ where $\lambda$ indicates the circular polarization (about $\mathbf{k}$). We have that ...
3
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2answers
264 views

Where does this term “shell” with prefix “on-/off-” come from?

Is there some historical reasons or is there a specific reason behind it? This question is connected to: Why on-shell vs. off-shell matters?
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15 views

Entropy of Reeh-Schlieder correlations

Any state analytic in energy (which includes most physical states since they have bounded energy) contains non-local correlations described by the Reeh-Schlieder theorem in AQFT. It is further shown ...
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+50

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 ...
3
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1answer
90 views

Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
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37 views

Is it possible to define a notion of temperature in a microcanonical ensemble?

I am thinking of a mircrocanonical ensemble as a finite system for which the number of particles, volume and the total energy is fixed. Is there a more refined view of this? Can I think of ...
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1answer
48 views

$\mathrm{d} \Omega_{CM}$ for a $1\rightarrow 2$ particle decay?

The differential solid angle is described in e.g. Srednicki's QFT text but only for the case of scattering. Because in the case of scattering it's defined with respect to the incoming three-momentum ...
2
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2answers
71 views

How to get conserved currents of a theory which are not Noether currents?

In the first SuSy lecture last week following theory of two real scalar fields has been considered as a first example: $$\mathcal{L}=(\partial_\mu \phi_1)^2/2+(\partial_\mu ...
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1answer
57 views

How use the Higgs branching ratio plot to extract information about the Higgs mass compared to experiment?

What does the plot of higgs branching ratio (see figure below) say about the higgs mass anyway? How can one use it as a guide to find the higgs mass experimentally? If we e.g. go to $M_H=126$ GeV ...
2
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1answer
32 views

All angle dependence in $\mathrm{d}LIPS_2$?

Recall that $\mathrm{d}LIPS_2$ (one particle decaying into two particles of the same mass) is given by $$\mathrm{d}LIPS_2 = \frac{\vert{\bf k_1'}\vert}{16\pi^2\sqrt{s}}\mathrm{d}\Omega_{cm}.$$ In a ...
3
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1answer
98 views

Mandelstam variables 1 positive 2 negative

The three Mandelstam-variables are defined as: $$s=(p_A+p_B)^2=(p_C+p_D)^2,$$$$t=(p_A-p_C)^2=(p_B-p_D)^2$$$$u=(p_A-p_D)^2=(p_B-p_C)^2.$$ Where A and B are the incoming particles and C and D are the ...
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46 views

One question about Weinberg's derivation of arbitrary spin fields expressions

In his book "QFT" (vol. 1) Weinberg writes the expression for an arbitrary spin massive field: $$ \hat {\Psi}_{a}(x) = \sum_{\sigma = -s}^{s} \int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2 ...
2
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0answers
44 views

Epsilon Tensor in FeynCalc

A few days ago I started to use the Mathematica package FeynCalc and one thing confuses me: Assume we have a four-vector $p_\mu$ and we contract it with the epsilon tensor. FeynCalc produces ...
3
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1answer
53 views

Dirac operator Feynman propagator

Is it true that the following identity holds for the Feynman prescription Dirac propagator: $$ S_F(x) \stackrel{?}{=} \gamma^0[S_F(-x)]^\dagger\gamma^0 $$ where $S_F$ is defined as the Green's ...
2
votes
3answers
171 views

why cannot fermions have non-zero vacuum expectation value?

In quantum field theory, scalar can take non-zero vacuum expectation value(vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory ...
4
votes
4answers
70 views

why do the electroweak vacuum have to be charge and color neutral?

My question is why the electroweak vacuum of the Standard Model have to electroweak charge and QCD color neutral? What goes wrong if electroweak vacuum has either non-zero charge or color quantum ...
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30 views

Regulating a particular function

I am interested in computing the integral of this function: \begin{align} \int_0^\infty\frac{2du(u^2+1)}{(1-e^{2\pi u})}, \end{align} which of course at first sight, does not converge. But in QFT ...
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1answer
36 views

can interaction between a massless fermion and external sourse exist?

For example, let's consider the electromagnetic interaction between a massless fermion and a electromagnetic externel sourse $A^\mu$, then the lagrangian is ...
4
votes
1answer
70 views

Lorentz transformations of fields evaluated at a point

I'm am sure that I must be missing something very simple, so apologies in advance. Considering the Lorentz transformation $\Lambda$ of a spinor fields, for the plane-wave solution $u(p)$, I cannot ...
6
votes
1answer
63 views

About the recent discovery of 4-quark boundstates.

I am referring to this, http://home.web.cern.ch/about/updates/2014/04/lhcb-confirms-existence-exotic-hadron So how does this work if we stick to keeping quarks in the 3 dimensional fundamental ...
0
votes
1answer
48 views

Fermion propagator is not a Grassmann-odd object?

Is the following differentiation correct: $$ \frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$ where $\eta$ is a Grassmann-valued ...
1
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1answer
34 views

Defining quantum effective action (Legendre transformation), existence of inverse (field - source)?

Given a Quantum field theory, for a scalar field $\phi$ with generic Action $S[\phi]$, we have the generating functional $$Z[J] = e^{iW[J]} = \frac{\int \mathcal{D}\phi e^{i(S[\phi]+\int d^4x ...
5
votes
1answer
51 views

$P$ symmetry that is apparent with one definition of fields but not with another

Suppose that we have a Lagrangian density like $$\mathcal L = -\frac{1}{4} \operatorname{tr} F_{\mu\nu}F^{\mu\nu} + \frac{\theta}{32\pi^2} \operatorname{tr} \big( \epsilon^{\mu\nu\rho\sigma} ...
0
votes
1answer
60 views

Is $\langle k \vert k_1k_2\rangle=0$

Using that $$ \vert k_1k_2\rangle = a^\dagger({\bf k_1})a^\dagger({\bf k_2})\vert 0 \rangle$$ and the commutation relations $$[a({\bf k}),a^\dagger({\bf k'})]=(2\pi)^32\omega\delta^3(\bf {k}- \bf ...
0
votes
1answer
38 views

Normalization of $\langle p_1 p_2 \vert p\rangle$ in RelQM and NonRelQM

Suppose a particle p of three momentum $\vec p$ decays into two particles of 3-momentum $\vec p_1$ and $\vec p_2$. I know the question might sound stupid but right now my brain is full stop: Is the ...
2
votes
1answer
32 views

A question about propagator of Maxwell field in different gauge

The propagator of Maxwell theory is different, depending on the gauge fixing procedure used. Then why will the S-matrix elements be the same for the same process in different gauges?
3
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1answer
59 views

References on $C^{*}$-algerbas, $W^{*}$-algebras and Quantum Theories

I would like to know some references regarding $C^{*}$ and $W^{*}$-algebras and quantum theories. I'm interested in concrete physical applications, models and problems. Here it is the list of ...
1
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1answer
68 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 ...
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0answers
72 views

Functional integral aproach for Feynman rules

I am familiar with the basic ideas of quantum field theory but I feel uncomfortable when I have to derive Feynman rules by myself for a given action (for example in non-linear sigma models or ...
4
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1answer
98 views

Anti-symmetric forms on Dirac spinors

In order to describe invariant forms on Dirac spinors $S$ one can find trivial subrepresentations in $S \otimes S$. If we use $S \cong (1/2, 0) \oplus (0, 1/2)$ then \begin{multline} [(1/2, 0) ...
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1answer
23 views

Why is the periodicity of fields in finite temperature QCD consequence of Trace in the action?

In finite temperature QCD, the gauge fields must be periodic in temporal direction. They say this is the consequence of trace in the action for gauge fields. How does trace imply that the fields must ...
4
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0answers
213 views

Nature of Microscopic space-time

I am going through the introductory chapter's of Schwinger's Source theory. He writes, It [Source Theory] is a phenomenological theory, designed to describe the observed particles. No speculations ...
2
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1answer
100 views

Connection between QFT and statistical physics of phase transitions

I have heard that there is a deep connection between QFT (emphasized by its path-integral formulation) and statistical physics of critical systems and phase transitions. I have only a basic course in ...
4
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1answer
76 views

Allowed interactions in bosonic string theory

In a quantum field theory, only a finite set of interactions are allowed, determined by the Lagrangian of the theory which specifies the interaction vertex Feynman rules. In string theory, an ...
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0answers
26 views

Understanding Bose enhancement in reheating

I'm struggling to understand the Bose enhancement in reheating. I've read that: At the end of inflation, the inflaton field, $\phi$, is something like a condensate with excitations of a single ...
1
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1answer
87 views

How to prove useful property of logarithm of generating functional in QFT?

How to prove that $\ln(Z(J))$ generates only connected Feynman diagrams? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point.