# Tag Info

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There are really several questions here: (a) What is the renormalization group? Specifically the law of composition, etc. (b) How does the equation the OP gave relate to this? Short Answer It's a semigroup (see references below). The equation you wrote, $$\tag{1} \left[\mu \frac{\partial}{\partial \mu} + \beta \frac{\partial}{\partial g} + m \gamma_{m^2} ... 7 You are right, it is wrong to think that in gauge theory "gauge transformations are just a redundancy". This becomes true only if one abandons locality, ignores all boundary effects, all instanton effects, hence most of what is interesting about gauge theory. Of course forming gauge equivalence classes (say of observables) is something one wants to do every ... 5 It's not a sufficient explanation. There are asymptotically free theories which are not strongly coupled in the IR. The rate at which the coupling gets strong is important. In QCD, it seems to get strong very quickly near the confinement scale, so that beyond a certain scale, you only see hadrons. It is not really understood how this works. The ... 4 To be honest, I think that the route you describe (and which is also used in many textbooks) is not physically well motivated at all. You have begun with a theory of a fermion with a global symmetry which maps physical states to different physical states. This theory has the property that specifying initial conditions on a spacelike surface completely ... 4 I think you misunderstood what the professor wanted to say. To understand this, let us evaluate the integral more thoroughly (your expressions contain some mistakes). If we use the dimensional regularization prescription d\rightarrow d-2\epsilon and an additional mass scale \mu, we get for the integral in question the following result:$$\int ...

3

$|\Omega\rangle$ is the vacuum of the full interacting theory and $|0\rangle$ is the vacuum of the free theory. They are related in the following way $$|\Omega\rangle = \lim_{T\rightarrow(1-i\epsilon)\infty} (e^{-iE_0(T+t_0)}\langle \Omega|0\rangle)^{-1}e^{-iHT}|0\rangle$$ and the correlators $$\langle \Omega|\phi(x_1)\phi(x_2)...\phi(x_n)|\Omega\rangle = ... 3 The connection of superconductivity to Seiberg-Witten theory can be understood through the observation that superconductivity is related to the Meissner effect, which is the exclusion of magnetic field lines from a superconductor. Seiberg-Witten theory is based on the analysis of the moduli space of an \mathcal{N}=2 supersymmetric Yang-Mills theory. It ... 3 (1) Classifying "Phase Structure of (Quantum) Gauge Theory" (with a gap) is about the same as classifying phase structure of topologically ordered states. Some topologically ordered states are described by a group and can be related to a gauge theory. Some other topologically ordered states are not related to gauge theory. (2) One way to classify "Phase ... 3 Not all irreducible representations (irrep's for short) of the Poincaré group lead to a Lagrangian. One example (see my comment to Julio Parra's answer) are the zero-mass, "continuous-helicity" (sometimes called "infinite-helicity") representations. There is, however, a way to begin from a positive energy irrep of the Poincaré group (i.e. a 1-particle ... 2 Under a Wick rotation, which is what you do in order to go from Minkowski to Euclidean space, both the partial derivative \partial_0 with respect to time and the zero-component of the gauge field transform as$$\partial_0\rightarrow i\partial_\tauA_0\rightarrow iA_0.$$This defines the covariant derivative for statistical field theory. 2 In principle one has to calculate the pole of correlation functions involving gauge invariant operators like \text{Tr}F_{\mu\nu}F^{\mu\nu}. The problem is that due to asymptotic freedom, QCD is not solvable perturbatively at low energies. This is why nonperturbative techniques like lattice QCD are used to calculate such spectra. A key achievement in this ... 2 I have to admit that I have no idea about the model you are working on, but the standard way to determine whether a gauge theory is confining or not is to calculate the vacuum expectation value expectation value of Wilson loops. The latter are gauge invariant operators that describe parallel transport around a closed loop in spacetime. If the vacuum ... 2 I believe in the last line, the plane-wave functions u_k(x) should carry different coordinates and momenta, e.g$$ [a(k)^\dagger,a(k')]u_k(x)u_{k'}(x') $$You may note that the commutator [\phi(x),\pi(x')]=i\hbar\delta(x-x') holds if one choses [a_k,a_{k'}^\dagger]=\delta_{kk'}. However, this indirect reasoning is no proof that this choice is unique. ... 2 I'm not sure such a thing exists. Usually reps only helps you classify the kind of particles you have (i.e the quantum numbers that identify them) and how they transform under the corresponding group. I believe how to represent this particles mathematically and what is their dynamics is a different matter. The only thing similar I know about is that some ... 2 I think you're confusion is with notation since unfortunately, two notation often used to denote projected spinors. One notation is to write: $$\psi \equiv \left( \begin{array}{c} \psi _L \\ \psi _R \end{array} \right)$$ In this notation  \psi _L  and  \psi _R  are two component Weyl spinors. However, a second notation ... 1 What is the usefulness of the Majorana representation? Majorana spinors are used frequently supersymmetric theories. In the Wess-Zumino model - the simplest SUSY model - a supermultiplet is constructed from a complex scalar, auxiliary pseudo-scalar field, and Majorana spinor precisely because it has two degrees of freedom unlike a Dirac spinor. The ... 1 In the linear sigma model, the chiral action on the pion fields can be implemented on the following matrix combination of the fields:$$U(2) \ni \Sigma = \sigma + i \tau^a \pi_a $$An element  (U_L = exp(\frac{i}{2}\theta^{(L)}_a \tau^a), U_R = exp(\frac{i}{2}\theta^{(R)}_a \tau^a)) \in SU(2)_L \otimes SU(2)_R  acts on \Sigma as follows:$$\Sigma ...

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This is how I understand this issue. First, I believe you may agree that imposing gauge invariance is a sensible thing to do. If we want our fields to be invariant under some kind of transformation it better be local, since two separate space-time points shouldn't be related in any unnecessary way, otherwise we may violate causality. A different issue is ...

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Alex Nelson's answer is much better that mine, but it doesn't address your question at all. (a) Does it form a group? No, it doesn't. See bellow, to find out what does 'look like' (but isn't) a group. (b) What are the elements of the group? The group-like structure is the following. Being sloppy, effective action satisfies the folowing semigroup ...

1

Let us quickly run through the standard KK compactification. We start with a $d+1$ dimensional theory $$S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x \sqrt{G} R_{d+1}$$ More general actions on the $d+1$ dimensional space can be considered, but this will suffice for our purposes. The metric $G_{MN}$ can be decomposed as  ds^2 = G_{MN} dx^M dx^N = e^{2\Phi} ...

1

As previous answers have correctly noted gamma matrices do not forma a basis of $M(4,\mathbb{C})$. Nevertheless you can construct one from them in the following way 1 the identity matrix $\mathbb{1}$ 4 matrices $\gamma^\mu$ 6 matrices $\sigma^{\mu\nu}=\gamma^{[\mu}\gamma^{\nu]}$ 4 matrices $\sigma^{\mu\nu\rho}=\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}$ 1 ...

1

The symmetry factor should be $2$. This comes from the fact that exchanging the derivatives at the vertex is the same symmetry operation as swapping the endpoints of the propagator in the loop. Each of them amounts for a multiplication by 2, but since they are identical we are essentially overcounting. Dividing by $2$ corrects this error.

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You might try it by the standard way of calculating Green's function (which is what the propagator is) of linear differential operators. For this purpose you have to derive the equations of motion from the given Lagrangian, which will be some operator acting on the gauge field. The crucial observation is now that the Green's function can be calculated by ...

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The statement you cited does not imply that a complex representation of a gauge group implies a chiral gauge theory in general. This only holds true if the gauge group corresponds to a chiral symmetry in the first place. A chirally symmetric theory contains massless fermions. Regarding your counterexample: it is true that QCD contains fermions in the ...

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What is really being said is that it is not an uncaused event in the vacuum, but in simplistic terms, it could be said to be more a property of the vacuum. No event is taking place as such. Describing it as an event is trying to put it in simplistic terms, which while it may help in understanding to some degree, is not an exact description. You could equally ...

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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 ...

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