# Tag Info

3

The existence of auxiliary fields can be motivated in two different (but related) ways that I know about. The first is using superspace. Superfields are functions of position and two Grassman variables. The auxiliary fields are necessary terms to ensure that a superfield remains a superfield under SUSY transformations. The second way to motivate auxiliary ...

0

Maybe, it will be particularly the answer on your question. It's convenient to classificate the fields due to Wigner classification of the Poincare group representation. First assume only massless case. In this case there aren't mass Casimir operator $\hat {P}^{2}$ and spin Casimir operator $\hat {W}^{2}$, but the Pauli-Lubanski operator is proportional to ...

2

The lagrangian of the gauge field is independent of that of the scalar field. You have to "guess" it. The reason we pick this one is twofold: 1-it is the one which gives the Maxwell equations, so when you try to describe E&M, that looks like a good guess; 2- if you think of all the terms that are both Lorentz invariant, parity invariant and gauge ...

6

Let's compare classical mechanics and GR to attempt to get at the intuition you're looking for. Classical mechanics. Recall that in the classical mechanics of a system of $N$ particles, the configuration of the system at every point in time is represented by a point $x\in\mathbb R^{3N}$. The configuration manifold $\mathcal Q$, namely the set of possible ...

0

Full field theories generally take up all of space. However, occasionally it is more convenient to work with sums instead of integrals. This can be done by artificially putting the system in a box. But when putting a system in a box you get edges and just like in any other theory you need to give the boundary conditions (the same is true for example in ...

0

Firstly, you ask isn't there more than one symmetry meaning this question has several answers? Yes! In general, a given theory can have all sorts of symmetries, and each of these symmetries leads to its own conserved quantity via Noether's theorem. As for what's going on with Noether's theorem and applying it in general, I'd like to strongly ...

3

The equation of motion corresponding with $\mathcal{L}_{kin}$ is $$(∂_{t}^2-{∂_{\mathbf{x}}}^2)φ=0,$$ the Klein-Gordon equation, which has its origin in relativistic field theory. The minus sign is essential for relativistic invariance and leads to propagating solutions (waves). With$$φ(\mathbf{x},t)=\text{exp}[iωt]ψ(\mathbf{x}).$$ we ...

5

Think of a two dimensional horizontal elastic sheet in three dimensions. Suppose we specify the height everywhere with $\phi(x,y)$. Then if the sheet moves up and down there is kinetic energy. This kinetic energy is proportional to $\dot{\phi}^2$ because $\dot{\phi}$ is telling you the velocity of that point on the sheet. Also, suppose we want to pull ...

3

First let's rewrite the kinetic term $\mathcal{L_{kin}}$ such that the metric $\eta^{\mu \nu}$ is explict $$\mathcal{L_{kin}}=\frac 12 \eta^{\mu \nu}\partial_\mu \phi\partial_\nu\phi$$ In this answer we will restrict our attention to 4-dimensional Minkowski space-time $M^4$, the flat "arena" of special relativity, and set the speed of light $c=1$. The ...

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To understand the difference between the Higgs mechanism and the mass generated by a Chern-Simons term one has to realize that the Higgs case is related to a (spontaneously) broken symmetry. Assume that you have written down a field theory which admits a set of equivalent ground states (vacua). These states can be transformed into each other by continuous ...

2

In classical field theory, the system will indeed be in the minima of the potential, i.e., the point at which $\partial V/\partial \phi_i=0$ for all fields $\phi_i$ for all fields $\phi_i$ (more precisely one should include the fermionic fields here as well but fermions don't exist in a classical world). In Quantum Field Theory the same applies as above ...

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Scale invariance refers to invariance under scaling the coordinates i.e., $x^\mu \rightarrow \lambda\ x^\mu$ ($\mu=0,1$ in this case). One needs to associate a (naive) scaling dimension to the fields -- this is done as follows. Suppose that $$\psi(\lambda\ x) = \lambda^\Delta\ \psi(x)\ .$$ Plug this into the action and use the kinetic term to figure out a ...

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