8 votes
Accepted

Similarity transformations in QFT

I would like to give you a very important example related with the comment posted by Connor Behan. In the standard model Lagragian we find out that diagonalization is not possible for all terms in the ...
  • 316
4 votes
Accepted

What's the difference between the conjugate momenta in the classical mechanics and in field theory?

In field theory with Lagrangian $$\begin{align}L&[q(\cdot,t),v(\cdot,t);t]\cr ~=~&\int \!d^nx~{\cal L}\left(q(x,t),\partial q(x,t),\partial^2q(x,t), \ldots;\right. \cr &\left. v(x,t),\...
  • 175k
2 votes
Accepted

Clarification for derivatives under a change of variables

If $f(x,t) = f(x + ct)$, then the chain rule gives \begin{align} \frac{\partial f}{\partial x} = \frac{\partial f}{\partial (x+ct)}\frac{\partial (x+ct)}{\partial x} = \frac{\partial f}{\partial (x+ct)...
  • 6,712
2 votes

Doubt in classical field theory/electromagnetism

An electromagnetic field is generated by some distribution of charges and currents. The electric field tells you what force would be exerted on a “test charge” at a point in space and time. The ...
1 vote
Accepted

Integration with the Dirac delta function

Integration variables can be renamed at will: $$ \int dr_{i} e^{-\mathrm i \beta q \phi(r_{i})} = \int dr e^{-\mathrm i \beta q \phi(r)} . $$
1 vote
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Peskin and Schroeder's QFT eq. (9.14): Gaussian momentum field integration of phase space path integral

Briefly, there are 2 issues: It is safest to Wick rotate $t_E=it_M$ to make the Gaussian integrals exponentially damped rather than oscillatory. (NB: Don't also Wick rotate the momentum field $\pi_M=...
  • 175k
1 vote
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Relativistic invariants of a classical field in 4D fashion: why the relation between the components of the current density holds?

The definition of $dS^i$ (Landau §6) is that it is a four-vector equal in magnitude and normal to the hypersurface element; in other words, $dS^i$ is the projection of the hypersurface element, ...
1 vote
Accepted

Scale factor in conjugate scalar field inside conformally flat spacetime

This is why I prefer writing down the action of your field theory, which reads $$S = \int d^4x \sqrt{-g}\mathscr{L}$$ One should better first simplify everything taking into account $\sqrt{-g}$ and ...
  • 849
1 vote
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Meaning of Bogolyubov transformations

This answer is inspired by the approach taken in Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. When you canonically quantize a theory, you are typically trying to find ...
1 vote

Meaning of Bogolyubov transformations

I'm not sure how much this high-level overview will help on its own, but maybe the material I've linked to will be a good starting point. transform even in flat spacetime If you describe Minkowski ...
  • 22.9k
1 vote

Regarding notation used for infinitesimal parameters of the Lorentz algebra and generators of the Lorentz group

As I suspect many students of field theory will have the same question, I'll elaborate on @Rindler98 's answer: In the defining representation of the Lorentz group the matrix ${\omega^\mu}_\nu$ is ...

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