4 votes
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Pauli-Lubanski vector for Maxwell's equation

I will attempt to give an answer to your question and if there are any points for which something is not clear, please let me know in the comments... First of all, we shall note that under ...
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4 votes

How is the Feynman propagator (Green's function) connected with the field?

The Feynman propagator is the time-ordered two point correlation function of the field \begin{equation} \langle 0 | T\phi(x)\phi(y) | 0 \rangle = D_F(x,y) \end{equation} Because $D_F$ obeys the ...
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3 votes

How is the Feynman propagator (Green's function) connected with the field?

You were right in your comment that what I had suggested, i.e. choosing $u(x)=\phi(x)$ was trivial and could not be used to infer any useful results. A more insightful choice would be $f(y)=\phi(y)$. ...
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3 votes
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Equal-time Canonical Commutation Relation for a scalar field

You can write the integrals in question as $$\int_{\mathbb{R}^3} f(\vec{p}) d^3 p = \int_{-\infty}^\infty\!\!\int_{-\infty}^\infty\!\!\int_{-\infty}^\infty\!\! f(p_x,p_y,p_z) dp_x dp_y,dp_z$$ where ...
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3 votes
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A wind tunnel and 2 strong magnets in the wind tunnel creating a very strong field, how would the wind & magnetic force interact?

Magnetic fields exert a force on charged objects and magnetic objects. Air is usually neither. If a charged particle moves in a magnetic field it experiences a force proportional to the charge, field ...
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2 votes

What is the problem with classical fermionic field?

Note that the soul-part of a supernumber [and in particular a Grassmann-odd variable] is an indeterminate/a placeholder/has no value. This is fine as long as we make manipulations within the framework ...
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Calculation of Lagrangian from Hamiltonian $\frac{1}{2}(-i\partial_\phi -A)^2$

We start with the Lagrangian $$ L_M~=~\frac{m}{2}\left(\frac{d\vec{r}}{dt_M}\right)^2 + q \vec{A}\cdot\frac{d\vec{r}}{dt_M}-q\phi_M, $$ for a non-relativistic point particle in Minkowski space ...
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2 votes

Help with an integral in Peskin & Schroeder - QFT

Hint: $$\begin{align} \int_{m}^{\infty}\! dE~\sqrt{E ^2 - m ^2} e ^{-iEt} ~=~~~&\left(\int_{m}^{-i\infty} +\int_{-i\infty}^{\infty} \right)\! dE~\sqrt{E ^2 - m ^2} e ^{-iEt}\cr ~\stackrel{E=|E|e^{...
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2 votes
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Conjugate momentum for constant scalar field

I would like to try and answer my own question, which I was able to do taking hint from Connor's answer. The essential point is that the differential equation $f''(x) = 0$ has a qualitatively distinct ...
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2 votes

Conjugate momentum for constant scalar field

I haven't gotten around to reading that paper yet but zero modes are not pure numbers. Before we quantize a free scalar field in a finite box, we write it as \begin{equation} \phi(\textbf{x}, t) = \...
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2 votes

Field shift in free Klein-Gordon theory

$\newcommand{\ip}[1]{\left<#1\right>}\newcommand{\d}{\mathrm{d}}\newcommand{\inv}[1]{{#1}^{-1}}$What happens here is a simple completion of a square. To see it in its cleanest form, without ...
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2 votes
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How do you multiply a 4x1 spinor by a $SU(3)$ matrix?

Color and Dirac/spinor indices are different types of indices. E.g. the quark field carries both.
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1 vote
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Deriving equation describing fermion-antifermion field

No, the first equation you wrote already describes particle and anti-particles. This is what puzzled Dirac the most and what ultimately gave him his Nobel. To understand this note first that the field ...
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1 vote

Noether charge on complex scalar field

Well, basically you ignore the anti-symmetry of the Pauli matrices. For example, since $i$ and $j$ are dummy indices, I can perform the substitution $i\leftrightarrow j$ and hence starting from the ...
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1 vote
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Why Electron Quantum Field Wants Little Energy But Photon Field Doesn't

When we are calculating the scattering probability for particles we can do this using Feynman diagrams. For example the tree level diagram for electron positron scattering to two photons is: (diagram ...
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1 vote

Why Electron Quantum Field Wants Little Energy But Photon Field Doesn't

Be careful of metaphorical statements like the electron field "wanting" something. The electron-photon system described by QED has an interaction where either an electron can radiate a ...
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1 vote

What's the difference between these two lagrangian?

The Lagrangian (2) can always be rewritten into the form of the Lagrangian (1) [with possibly a different functional dependence]. The other way requires some geometric input, i.e. a connection $\nabla$...
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1 vote
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How to properly get low energy effective field theory of superfluid?

$$ {\cal L} = \varphi^\dagger(i\partial_t +\frac 1{2m} \nabla^2 +\mu)\varphi -\frac \lambda 2 (\varphi^\dagger\varphi)^2. $$ Here $\mu$ is the chemical potential for the bosons. The potential part of ...
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1 vote

Why does the Lagrangian have $O(4)$ symmetry after Wick rotating (previously Lorentz symmetry)?

As long as the Minkowski action is constructed from Lorentz-covariant tensors, then under Wick rotation [where the contravariant and covariant $0$-components of the tensors are Wick-rotated in ...
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1 vote
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Expression of a Lagrangian in other form

What you need to use are so-called null-Lagrangians, i.e. terms that are a divergence and can be dropped in the Lagrangian because in the action integral they only contribute a surface term, or, to ...
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