# 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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### Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask me ...
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### What is a field, really?

There was a reason why I constantly failed physics at school and university, and that reason was, apart from the fact I was immensely lazy, that I mentally refused to "believe" more advanced ...
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### Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
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### Deriving Lagrangian density for electromagnetic field

In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form $$\mathcal{L} =-\frac{\epsilon_0}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$ and ...
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### Why don't people use Hamilton's equations for a relativistic free particle?

A relativistic free particle has the Hamiltonian in general: $$\mathcal{H} = \sqrt{{\bf p}^2c^2+m^2c^4}.$$ I read somewhere that says, it is possible to go further and say that the EoM are Hamilton's ...
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### Squaring the E&M (Maxwell) field strength tensor

In Section 3.4 of Schwartz's "Quantum Field Theory and the Standard Model", the square $F_{\mu\nu}^2$ of the field strength tensor $F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu}A_{\mu}$ is ...
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### Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
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### What is meant by a local Lagrangian density?

What is meant by a local Lagrangian density? How will a non-local Lagrangian look like? What is the problem that we do not consider such Lagrangian densities?
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### Form of the Classical EM Lagrangian

So I know that for an electromagnetic field in a vacuum the Lagrangian is $$\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu},$$ the standard model tells me this. What I want to know is if there is an ...
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### What canonical momenta are the "right" ones?

I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the ...
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### Simple conceptual question conformal field theory

I come up with this conclusion after reading some books and review articles on conformal field theory (CFT). CFT is a subset of FT such that the action is invariant under conformal transformation ...
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### Beyond Hamiltonian and Lagrangian mechanics

Lagrangian and Hamiltonian formulations are the bedrock of particle and field theories, produce the same equations of motion, and are related through a Legendre transform. Are there more such ...
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### Conserved charges and generators

For the Klein Gordon field, the conserved charge for translation in space is given by: $$\vec{P}=\frac{1}{2}\int d^{3}k \, \vec{k}\{a^{\dagger}_{k}a_{k}+a_{k}a^{\dagger}_{k}\}$$ If we were to find ...
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### Difference between field and wavefunction

Can someone give me a clear explanation of what is the difference between a classical field, a wave function of a particle and a quantum field? I haven't find a clear explanation. For example for ...
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### When is numerical value of Lagrangian evaluated on-shell a full differential?

I noticed recently that for many field equations, Lagrangian evaluated on-shell (i.e. using equations of motions) is a full derivative- a divergence or something, or in other words a boundary term. ...
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### Need for a side book for E. Soper's Classical Theory Of Fields

I am reading now E. Soper, Classical Theory Of Fields, now and sometimes it is very hard to follow the equations. So I need a side book on classical field theory to read it comfortably. Landau & ...
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### Why can't General Relativity be written in terms of physical variables?

I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically ...
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### Examples of "gauging a global symmetry"

I am looking for someone to exemplify the actual process of "gauging a global symmetry." I am familiar with gauge bosons, gauge theories (QED), and the definition of "gauging a symmetry" etc., but I ...
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### Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
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### Showing that Coulomb and Lorenz Gauges are indeed valid Gauge Transformations?

I'm working my way through Griffith's Introduction to Electrodynamics. In Ch. 10, gauge transformations are introduced. The author shows that, given any magnetic potential $\textbf{A}_0$ and electric ...
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