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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Diagrammatics of Slavnov Taylor Identity

Is there a reference other than the original paper of 't-Hooft and Veltman, where I can get a pedagogical introduction to the diagrammatic approach to understanding the BRST-Ward or Slavnov-Taylor ...
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143 views

How to compute this loop integral? [closed]

I have a gauge boson that splits into two scalars and the loop is closed by a gauge boson as shown in the picture. The incoming boson has $\mu$ index while the boson that runs in the loop has momentum ...
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53 views

Basic QED - How are conserved charges expressions throught ladder operators derived?

I can't find this in similar questions, and I must be missing something very basilar since I can't find this in any textbook or online note: they just skip the passage. So, from my course's notes, we ...
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Definition of Fermion [closed]

Recently, I encounter a problem about the definition of Fermion operator. In our standard textbooks, the Fermions are defined by their exchange/braiding property, that is, if a minus sign appears by ...
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76 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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The Dirac equation for helium?

How to write down the Dirac equation for the two electrons in the helium atom? The problem is the interaction term, as $1/|r_1 - r_2|$ is apparently not Lorent-covariant.
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57 views

Are there any tests of quantum field theory one can do using everyday objects?

One of the reasons I love physics is because many of the theories I can test using everyday objects around me. For example I can predict how long it would take for me to drop the ball of a roof using ...
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51 views

Vacuum Structure of Schwinger Model

Quantum Electrodynamics in one-space and one-time dimensions ($QED_{1+1}$) for charged fermions is called the Schwinger model. If the charged fermion is massless, then the model is called the massless ...
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41 views

Eigenvalue for interacting Hamiltonian [closed]

Consider the Hamiltonian $$H=\omega_{1} a_{1}^\dagger a_{1}+\omega_{2}a_{2}^\dagger a_{2}+\alpha a_{3}^\dagger a_{3}(a_{1}^\dagger a_{2}+a_{2}^\dagger a_{1})$$ with $$ ...
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157 views

effect of a simultaneous local and a global $U(1)$ symmetry breaking

EDIT : I am trying to figure out the effect of symmetry breaking in a $U(1)_Y\times U(1)_Z$ invariant lagrangian where $U(1)_Y$ is local symmetry of the Lagrangian and $U(1)_Z$ is a global symmetry of ...
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38 views

Is there a scalar field that is not a lorentz scalar if we begin with Lorentz invariant Lagrangian?

In Quantum Field Theory by Mark Srednicki chapter 3 and 4, he constructs Lorentz invariant theory for scalar field by assuming that the scalar field transforms by ...
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What is the missing step in this result regarding the creation operators in Fock space?

In the above extract from Simons and Altman: Condensed Matter Field Theory, I am having trouble getting from (2.3) to (2.4) in the case of Fermions (ζ=-1 and the n(subscript i) values are modulo 2). ...
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91 views

Embedding of particles into fields

For the classification of particles (Wigner 1939), we look for unitary representations of the Poincaré/Lorentz group. There are are only infinite-dimensional (non-trivial) unitary representations! To ...
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82 views

How is the functional integral over momentum performed in the case of the real scalar field?

Let's follow Peskin and Schroeder section 9.2, page 282. The Hamiltonian of a free real scalar field is $$H=\int{}d^3x[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+V(\phi)]$$ so the expression for ...
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72 views

In the context of quantum field theory, what does it mean to “couple” something?

Suppose I have the following Lagrangian density \begin{equation} \mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} \end{equation} The lecture notes I an reading suggest if I want to "couple to ...
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46 views

Understading triplet Majoron model

In the Higgs triplet Majoron model, the spontaneous breakdown of ungauged lepton number gives rise to two Numbu-Goldstone bosons. But isn’t the SU(2) symmetry also broken? I mean when the neutrak ...
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77 views

Problem with determining number of goldstone bosons

Consider a theory $$\mathcal{L}=(\partial_\mu\Phi^\dagger)(\partial^\mu\Phi)-\mu^2(\Phi^\dagger\Phi)-\lambda(\Phi^\dagger\Phi)^2$$ where $\Phi=\begin{pmatrix}\phi_1+i\phi_2\\ ...
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34 views

Klein Gordon equation in de-sitter spacetime with time dependent Hubble parameter

If i try to solve Klein-Gordon equation for a scalar field in de-sitter background, the usual method is to transform to conformal spacetime : $$ds^2 = -dt^2 + e^{Ht}\bf{dx}^2$$ $$=>ds^2 = ...
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Vortex-domain wall co-excitation

Both vortices (or disclinations) and domain walls are possible topological defects in a spin system with frustration, but I did't find reference about the interaction of these two. Do any stackers ...
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55 views

Where in nature does a fermionic coherent state occur?

We see evidence of bosonic coherent states everywhere. Lasers and microwave circuits naturally condense into photonic coherent states and resonators do the same except with phonons. A coherent state ...
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77 views

Propagator for massless spin 2 particle

In my quantum field theory class, we saw ad derived the propagator for both spin-0 and spin-1 particles, massless and massive. I am curious to know what the propagator looks like for a spin-2 ...
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56 views

How do quantum fields really couple?

The term "coupling" between quantum fields refers to certain terms in the Lagrangian (density) $\mathcal{L}$ where the respective field operators appear together, e.g. $g\phi^\dagger\psi $ with ...
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142 views

What is the connection between geometry of physical space and Hilbert space?

In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ ...
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271 views

Noether currents in QFT

I am trying to organize my knowledge of Noether's theorem in QFT. There are several questions I would like to have an answer to. In classical field theory, Noether's theorem states that for each ...
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41 views

If the given source is not conserved, then which gauge should we use in photon propagator?

The photon propagator in general gauge is $$D_F^{\mu\nu}=\frac{-g_{\mu\nu}}{k^2+i\epsilon}+\frac{\xi-1}{\xi}\frac{k^\mu k^\nu}{(k^2+i\epsilon)^2}.$$ In general textbook, the reason that the ...
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46 views

What does it mean to have a degenerate $S$-matrix?

The Coleman-Mandula theorem $D>2$ assumes that the quantum field theory may not have a degenerate $S$-matrix. But what does it mean to have a degenerate $S$-matrix? The $S$-matrix if I got it ...
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69 views

Weinberg soft photon integral

In deriving the rate of emission of arbitrary numbers of soft photons in a general QED process, Weinberg performs the following integral (equations 13.2.8-9): $$-\pi(\vec{p}_m\cdot ...
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118 views

Does the Lorentz invariance of equation of motion guarantee the Lorentz invariance of the solutions?

If I have a Lorentz invariant equation of motion, like Klein-Gordon equation, is the solution automatically guaranteed to be Lorentz invariant? I ask this question because of the discussion from Mark ...
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55 views

Is it possible to generalize quantum gauge theories? [closed]

I know that there are nonabelian gauge theories and their supersymmetric extensions. Mathematically, gauge theories basing on the fact that one can introduce a fiber bundle with a Connection. From ...
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152 views

Wess-Zumino Gauge in non-Abelian supersymmetric theory

I've got a question concerning non-Abelian supersymmetric gauge theories. Consider supersymmetric non-Abelian theory realized on chiral superfields $\Phi_i$ in a representation $R$ with matrix ...
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1answer
68 views

In what sense are photons emergent?

Recently I read in an essay by Wilczek: "Photons are mixtures of weak B3 and hypercharge C mesons. It is those objects, not the emergent photon, whose properties are ideally simple." Until now I ...
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24 views

2-body differential cross section in CM frame discrepancy

The standard equation for the 2-body differential cross section in the CM frame (from several references) seems to be: $$\frac{d\sigma}{d\Omega} = \frac{1}{64\pi^2s}\frac{q}{k}|\mathcal{M}|^2,$$ where ...
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Where do pions go in the spontaneous symmetry breaking of the linear sigma model?

I have a few questions to figure out Peskin 4.3 problem which is Linear sigma model about the interactions of pions at low energy. This model consist of N scalar fields governed by the Hamiltonian ($ ...
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1answer
38 views

Volume factor in Faddeev-Popov quantisation

In Faddeev-Popov quantisation, why does the integral over gauge parameter cancel the volume factor of the gauge group that's in the denominator? In fact, I don't understand where the volume factor ...
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105 views

Given a QFT Hamiltonian, is there a unique Lagrangian?

Consider a QFT in one spatial dimension specified by the following Hamiltonian density: $\mathcal{H} = -i \phi^\dagger \frac{\partial}{\partial x} \phi + V(\phi^\dagger,\phi)$ where $\phi$ is a ...
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75 views

Why would it transform like this under chiral symmetry?

Why would a $2 \times 2$ matrix of spinless fields $\Sigma$ transform as follows under the chiral symmetries? $$\delta \Sigma = i \epsilon_{L} T_a \Sigma - i\Sigma \epsilon_RT_a$$ Primary written on ...
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75 views

“Find the Lagrangian of the theory”

I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances ...
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Where does the $\gamma_5$ here come from?

If we have that $$\delta \psi_L = i \epsilon_L^aT_a\psi_L$$ and $$\delta \psi_R = i \epsilon_R^aT_a\psi_R$$ And then we say that the above can be written in terms of $\epsilon^a$ and $\epsilon^a_5$ ...
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97 views

Can we measure the electron spin independently of its magnetic moment?

What experimental evidence do we have for the intrinsic angular momentum of the electron (its spin)? I am specifically interested in whether we have a value for this that is independent of the ...
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288 views

The concept of particle in QFT

I never learnt QFT and I apologize for my (probably) elementary question. Somebody told me that in QFT a particle is viewed as an irregularity in the field. On the other hand, in an article in ...
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56 views

Relation between representations/classifications

Generally a quantum system can be characterized in the following way: its states form a representation space for every symmetry group of that system. The representation has to be unitary (or ...
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108 views

How can gauge invariance be unphysical?

Gauge symmetry is said to be "unphysical" because the transformations - unlike changes of reference frame - do not correspond to real physical operations. But the consequences of gauge symmetries are ...
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78 views

What is “momentum density” and why it important to QFT?

I am reading Quantum Field Theory for the Gifted Amateur. On page 98, they provide a summary of a basic canonical quantization procedure: Step I: Write down a classical Lagrangian density in ...
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2answers
361 views

Feynman paths of FTL velocity have imaginary momentum?

In this answer it is discussed that Feynman path integrals sums amplitudes for all possible paths, including those that are not time-like. If you take the momentum-space path integrals, I would ...
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1answer
66 views

what does Peskin's square root of a matric mean?

Peskin (Intro to QFT) is using the next symbols when discussing dirac fields - $\sqrt{p\sigma}$ with $\sigma = (1,\sigma^1,\sigma^2,\sigma^3)$ (unit & Pauli). For example he represents the dirac ...
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52 views

Off-shell legs in Feynman diagrams

I have a tree-level diagram with one leg being off-shell (its momentum beeing $\mathcal{O}(m_B)$). How do I treat this leg when computing the amplitude? Do I put in the propagator and ignore the ...
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138 views

How are the field operator and quantum state after a beam splitter and a polarizing beam splitter individually?

How are the field operator $\hat{a}$, $\hat{a}^\dagger$ and the quantum state (like coherent state $|\alpha>$, Fock state $|n>$) changed after a beam splitter and a polarizing beam splitter ...
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245 views

Does the fact that we cannot exactly solve the Standard Model undermine the validity of QFT?

I have seen discusstions of this types before: there is a question about photons or virtual particles or vaccuum, etc. And there is usually a good and clear explanation from the point of view of ...
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103 views

What assumptions about the action do we make or give up in transitioning from classical mechanics to quantum mechanics to quantum field theory?

I am reading Quantum Field Theory for the Gifted Amateur and I feel I don't have a good grasp as to how the Lagrangian and the action are used differently in (1) classical mechanics (2) quantum ...
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Do “typical” QFT's lack a lagrangian description?

Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. ...