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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Where do negative powers of $f_\pi$ in the hadronic amplitudes come from?

According to Peskin and Scrhroeder the pion decay constant $f_\pi$ is defined via the following matrix element $$\left\langle0|j^{\mu5a}(x)|\pi^b(q)\right\rangle=-if_\pi \delta^{ab} q^\mu e^{-iqx}$$ ...
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heavy quark pair spin states

Apparently, heavy quark pair can have "axial-vector" spin state, "vector" spin state, and two different "tensor" spin states. Can anyone explain to me what they are and why this is the case? Thank ...
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27 views

Quantising the magnetic monopoles the make Maxwell symmetric

I don't believe this has already been asked, but I might be wrong; sorry. One can add a magnetic charge density/magnetic monopoles to Maxwell's equations to make the theory symmetric between the ...
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24 views

Fourier transformation and mode expansions [duplicate]

Sorry as this is a rather trivial question, but I'm stuck with a certain implication. I'm working on exercise 1.7 from Polchinski where we are given an open string with boundary conditions ...
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36 views

How to find the remaining subgroup after some Higgs field gets a VEV?

Say we have a group $G$ and a set of Higgs fields in a representation $R$ of $G$. One of the Higgs fields in $R$ gets a VEV, how can I determine the remaining subgroup after this symmetry breaking? ...
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24 views

renormalization subtraction point, scaling

When we use minimal subtraction scheme, for instance, we have a dependence of coupling on a scale $\mu$. Using the $\beta$ function, we can observe the behavior of the coupling at different scale ...
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36 views

QED renormalization: mass and dirac field

Why the mass renormalization $Z_m$ and the field renormalization $Z_\psi$ in QED (MS-renormalized) does not contribute to the beta function computation? From Ward identity, I know that $Z_A=Z_e^{-1}$, ...
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Degenaracy in mass of $8$ and $27$ reps of $SU(3)$ in Coleman's Aspects of Symmetry

In Coleman's Aspect of symmetry he proposes an amusing problem in the first chapter. It asks us to consider a set of eight pseudo-scalar fields transforming in the adjoint representation of $SU(3)$. ...
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76 views

Fourier transformation and commutators

Sorry as this is a rather trivial question, but I'm stuck with a certain implication. I'm working on exercise 1.7 from Polchinski where we are given an open string with boundary conditions ...
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0answers
39 views

(Super)Gauge Fixing in Supersymmetry

I have three questions about gauge fixing in supersymmetry, one is general and the other two explicit: Why gauge fixing seems not important in supersymmetry? By "not important" I mean gauge fixing ...
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1answer
38 views

Expanding free scalar field in terms of ladder operators

I'm having some difficulty with the finer points of expanding a field in terms of ladder operators. Note that this is not identical to the other related question I asked. From Peskin / Schroeder; ...
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1answer
73 views

Textbook on QFT in curved space-time via path integrals

I am looking for an introductory textbook on QFT in curved space-time via the path integral method. I want to understand the following: How to build a generic perturbative QFT in curved space-time ...
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74 views

Particle annihilation - mathematical description, equations governing it? [duplicate]

I was wondering about this and I would like to know an explanation why do particles and antiparticles annihilate? I would be interested in phenomenological, but most importantly mathematic explanation ...
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71 views

What is the algebraic form of the momentum eigenstate?

I'm asking this in the context of trying to verify the equation $a^{\dagger}_{p} \vert 0 \rangle = \frac{1}{\sqrt{2\omega_p}} \vert p \rangle$. So far I have calculated $\vert 0 \rangle = ...
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41 views

Clarification on Use of Counterterms in Renormalized Perturbation Theory

In renormalized perturbation theory, it's unclear to me how exactly we add the necessary counter-terms. Do we: Draw all possible diagrams, including the diagrams of the counter-terms to some order ...
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1answer
80 views

Field expansion in Peskin & Schroeder

Peskin and Schroeder state something which I'm not fully understanding. More specificially I think it's just phrased in a way I'm not understanding. In the Schrodinger picture we can expand the real ...
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1answer
34 views

What do we take the functional determinant of in computing th effective action in the Background field method?

I have some schematic notes on computing the effective action and I would like someone to help me fill the gaps. We start with \begin{equation*} \int{}\mathcal{D}\phi\,e^{-iS[\phi]} \end{equation*} ...
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1answer
64 views

Non-relativistic limit of complex scalar field Lagrangian

I am trying to derive the non-relativistic Lagrangian for a complex scalar field from taking the non-relativistic limit of the complex scalar field Lagrangian. I am following the steps in "QFT for ...
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2answers
130 views

In QFT how do you write down the most general interactions?

This past year I took a QFT class and I now feel comfortable solving scattering problems, but I am still a bit perplexed by how physicists write down a Lagrangian in the first place. In particular, ...
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1answer
72 views

A trace formula of two noncommutative operators

In many cases of quantum many-body problems, the Hamiltonian $H$ can always be divided into two parts, i.e. $H_0$ and $H'$. In this occasion, one can systemically calculate the partition function ...
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24 views

Plane wave solutions of the Majorana equation

Let u(p) and v(p) be the plane wave solutions of the Dirac equation with positive respectively negative energy. In case of a solution of the Majorana equation the charge-conjugated solution is ...
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48 views

Choice of framing in Gravitational Chern-Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
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1answer
91 views

Doubts with basic renormalization

When we renormalize to obtain the physical mass, the $\Lambda$ dependence of the physical mass is removed by introducing the counterterms in the Lagrangian. So whether we put ...
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2answers
127 views

Gaussian integral on a Riemannian manifold

How do I estimate the Gaussian integral $\int d^nx \sqrt{g(x)}~e^{-x^2} $ on a Riemannian manifold $(M,g=det~g_{\mu\nu})$? I've tried to consider $\sqrt{g(x)}$ as an analytic function and expanded it. ...
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37 views

Dynamics and kinematics of quantum field theory

What is the difference between dynamics and kinematics of quantum field theory? I read that in QFT there is no possibility to keep the two things distinct because of a problem with the separability of ...
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1answer
29 views

Spin magnetic moment direction of a particle

Is the spin magnetic moment of a fundamental particle like an electron always aligned along the direction of the spin angular momentum (meaning that the magnetic moment and the spin operators have the ...
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2answers
108 views

Uncertainty principle in quantum field theory

Can the uncertainty principle be derived in quantum field theory? If yes, does is have a different interpretation than quantum mechanics because the coordinates $x_i$ are now parameters and not ...
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1answer
43 views

Why is cut-off regularization is not Lorentz invariant?

Why is it said that the cut-off regularization is not a Lorentz invariant regularization method?
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58 views

Violation of unitarity: meaning and consequences

What is meant by unitarity and violation of unitarity of a QFT? For example, Fermi theory of beta decay is said to violate unitarity. How does violation of unitarity make a theory sick?
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62 views

$\phi^4$ theory two-loop contributions

Wherever I see calculations of two-loop contributions to the $\phi^4$ propagator (such as Peskin, page 328, on the bottom), only the sunset diagram (aka the Saturn diagram) is considered, but not, ...
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35 views

A question about a consequence of symmetry in $\phi^4$ theory

Why does the symmetry $\phi→-\phi$ mean that an amplitude can be written as $\alpha+\beta p^2+\gamma p^4+...$ without the odd terms in $p$? I understand that, due to this symmetry, any diagram in ...
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1answer
117 views

Why can't quantum field theory be quaternion instead of complex?

So, the definition of QFT in terms of path integrals is that the partition function is: $$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$ But does it have any meaning if instead of this $U(1)$ ...
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1answer
59 views

What is the required differentiability of the solutions to do QFT?

Given a real scalar field satisfying: $$P\psi=(\square_{g}+m^{2})\psi=0$$ on a globally hyperbolic spacetime ($M,g_{ab}$). One can construct a $C^{*}$-algebra $A(M,g)$ ("the minimal algebra") which ...
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51 views

Why is the introduction of a quantization volume necessary for quantization of the EM field

I have been working through the quantization of the electromagnetic field, and every source I find introduces a quantization volume with periodic boundary conditions in the process, in which we fit ...
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101 views

Physical explanations for renormalization

Some related questions on Renormalization: Why is renormalization even necessary? My understanding is that the supposed problem is that the sums of certain amplitudes end up being infinite. But ...
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1answer
47 views

Wick's Theorem: Why is the vacuum expectation value of uncontracted operators zero?

I'm am right now reading Chapter 4.3 (Wick's Theorem) in Peskin & Schroeder. It is said that In the vacuum expectation value, any term in which there remain uncontracted operators gives zero ...
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1answer
42 views

Classical vacuum and quantum vacuum

How to determine the ground state of a classical field, for example an electromagnetic field? What is the difference between the the ground state of a classical field and that of a quantum field?
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1answer
51 views

QFT field expansion: Positive/negative energies and exponentials

I'm trying to understand the relationship between the exponentials in the field expansion and the energy of the of the particle being positive or negative. Is it necessary or is it a convention that ...
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1answer
71 views

Why is the chiral symmetry only $SU(3) \times SU(3)$ and not $SU(6)$?

In the limit where the masses vanish, low energy QCD has a well known chiral symmetry (see http://arxiv.org/abs/hep-ph/0505265 for a very extensive review, and pg 19 for the section relevant for my ...
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24 views

Why can we set mass to zero in Yukawa RGE derivation?

In problem 12.1 from Peskin&Schroeder's book I have to derive the beta functions in massless Yukawa theory. What's the justification for setting mass to zero and what's the difference between ...
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63 views

Goldstone's theorem massless states

I'm currently trying to wrap my head around Goldstone's theorem proof as given by Itzykson & Zuber. The reasoning is pretty straightforward, but there's a result that's bothering me from a ...
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1answer
78 views

Field theory in four dimensions

I was reading Schwartz's book on QFT. In chapter 14.5 at p.267, while speaking about path integral he says: [...] the path integral (and field theories more generally) is only known to exist (i.e. ...
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2answers
91 views

What role does “spontaneously symmetry breaking” played in the “Higgs Mechanism”?

In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of ...
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1answer
47 views

How is polarization vector in QFT related to polarization in classical electrodynamics?

As i know in classical electrodynamics polarization shows the orientation of the electric vector in a plane perpendicular to the direction of propagation of light. But in quantum field theory ...
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1answer
50 views

Compton Scattering Feynman diagram integral expression

I'm trying to write down the integral expression according to the feynman-rules for this Diagram of an electron with compton scattering and a one-loop correction: ![Compton Scattering][1] ...
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74 views

A classically charged point particle interacting with electromagnetism and gravity

Consider a classically charged point particle interacting with electromagnetism and gravity. The relevant dynamical variables are $\chi^\mu (\tau)$ of the particle, the electromangetic potential ...
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19 views

Pseudoscalar particle decay

Suppose I want to calculate amplitude of pseudoscalar particle decay into electron + positron. Interaction Hamiltonian is given by (ignoring the positive and real constants) $\mathcal{H} = \bar{\psi} ...
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14 views

Creating an arbitrary state of the quantum simple harmonic oscillator [duplicate]

Suppose $\mathcal{B}=\{\lvert 0\rangle, \lvert 1\rangle, \lvert 2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} ...
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28 views

Current density defined by the scattering operator

I have a problem with the definition of the current density. In most literature it is defined as $j^\mu=\frac{i}{2}(S^*\frac{\partial S(A)}{\partial A_\mu(x)})$. I understand that normally we use ...
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Analogy between a classical discrete system and non classical continous system

Most introduction textbooks about quantum fieldtheory start with a discrete classical harmonic oscillator and then looks at it in the continuous quantized case (quantized field). This leads to the ...