# Questions tagged [field-theory]

For questions where the dynamical variables are fields, that is, functions of several variables (typically, one time coordinate and several space coordinates). Comprises both classical field theory and quantum field theory. Use this tag when the question applies to both classical and quantum phenomena. Otherwise, use the specific tag instead.

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### Electric charges on compact four-manifolds

Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
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### Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
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### Connection between gauge invariance and Lorentz invariance

This question is presented in the context of Weinberg's QFT book treatment, in particular considering the electromagnetism chapter. It begins in chapter 5 where Weinberg argues that in order to have ...
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### Why is it that the equation of a massless scalar field *must* be conformal invariant?

I'm reading a paper [1], p.111 where it is said that: However, the equation of scalar field with zero mass must be conformal invariant while equation $\square\varphi=0$ does not satisfy this ...
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### Quantization during phase transition

Consider a scalar field $\phi(t,\vec{x})$ in $\mathbb{R}^{1,3}$ with the following lagrangian $$\mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - V(\phi)$$ where $V(\phi)$ is such that ...
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### How groups act on fields in QFT?

I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how ...
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### Non-linearities in Lagrangian of a scalar field coupled to point-like source

I have an exercise where I did not manage to understand the questions. Basically, I have this Lagrangian \mathcal{L}=\frac{1}{2}(\partial \pi)^2-\frac{1}{\Lambda^3}(\partial \pi)^2\...
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### Understanding the total spin as the Noether's charge and rotation generator of the Heisenberg model

Consider the Heisenberg model where the Hamiltonian $$H= J\sum_{\langle i,j\rangle}\textbf{s}_i\cdot \textbf{s}_j$$ has continuous rotational symmetry. Since $\textbf{s}_i\in\mathbb{R}^3$, the ...
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### Lorentz Symmetry Group as continuous symmetry for limit of discrete spacetime

There is a variety of models of quantum field theory, where discrete spacetime is used as technical support, or even suggested as physical reality. As far as I know, all of such models faced serious ...
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### Breakdown of the Legendre transform for the complex scalar field

Suppose we wish to obtain the energy density of the free complex scalar field $\varphi$ as a Legendre transform of the corresponding action. From Wikipedia, writing the action of a free complex scalar ...
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### Four vector potential and discrete parity transformations

I am having trouble understanding the effect of parity transformations on the four-vector gauge field (for example). I am working in three dimensions, but the analysis is probably not that different ...
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