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4

Note that the finite transformation of: $$ W^a_\mu \to W^a_\mu + \frac{1}{g} \partial_\mu \theta^a + \epsilon^{abc} \theta^b W^c_\mu $$ is: $$ W^a_\mu t^a \to g W_\mu^a t^a g^{-1} + \frac{i}{g} \partial_\mu g \tag{1} $$ where: $$ g = \exp(-i \theta^a t^a) \;\;\; \text{and} \;\;\; [t^a,t^b] = i \epsilon^{abc} t^c $$ Thus, the first term on the right-hand ...


6

Yes, you would have to introduce another gauge field. For example in the Standard Model there is gauge invariance under $SU(3)\times SU(2) \times U(1)$, and so there are three gauge fields: the gluons, the $W^\pm, Z$ weak gauge bosons and the photon. In general terms, it is simpler to argue like this: if you have gauge invariance under a Lie group $G$, the ...


2

Unification in physics is used differently in classical physics than in the quantum regimeof elementary particles. Unifying electricity and magnetism became necessary when functional measured relations appeared which connected the motion of charges with the magnetic field and the magnetic field with the motion of charges. The Biot-Savart law and Ampere's ...


1

The scalar potential and magnetic vector potential are combined into a four-vector, $A_{\mu}=(\phi,\vec{A})$ which is a gauge field, and in the language of differential geometry, a 1-form. The Lagrangian of the field theory (i.e. Maxwell theory) is, $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ where $F_{\mu\nu} = \partial_{[\mu}A_{\nu]}$ is the ...


2

I'll answer this question by example. Some standard gauge choices are the $R_\xi$ gauge and axial gauge with propagators $$ \Delta^\xi_{\mu\nu} (k) = - \frac{i }{ p^2 - i \varepsilon} \left[ g_{\mu\nu} - \left( 1 - \xi \right) \frac{ k_\mu k_\nu }{ k^2 } \right] $$ $$ \Delta^{\text{axial}}_{\mu\nu} (k) = - \frac{i }{ p^2 - i \varepsilon} \left[ g_{\mu\nu} - ...


0

I think because the wavefunctions are required to be normalized so that $\psi^{*}\psi$ represents the probability or probability density of finding the particle, so their amplitude are not allowed to scale arbitrarily. That's why the gauge field can only be real.


0

let's say the trace is the expectation value. the action will be invariant so by calculating the expectation value of the action one would expect a minima on the path taken by a particle. This would be independent of time, the same physics will describe the dynamics tomorrow. hence, periodicity in the temporal direction is a way of saying that if something ...


1

First of all there is no proof of this statement. It is just a general expectation that the more symmetries you have the more reason to expect better quantum properties. This works with SUSY, the more SUSY you have the better the theory is at the quantum level, say $N=4$ SYM, or $N=8$ SUGRA that some people still have hope to be well-defined. If you involve ...


2

This is a special case of a more general phenomena. Conserved currents never acquire anomalous dimensions, they are protected by the symmetry. If you have a conserved current, you have a symmetry algebra for the theory \begin{equation} [Q^a,Q^b]=if^{abc}Q_c \end{equation} For this to hold, the charges need to be dimensionless. But the charges are given by ...


2

On one hand, by including the Lautrup-Nakanishi field $B^a$, we have an off-shell BRST formulation, i.e. we can prove the nilpotency of the BRST transformation without using the (Euler-Lagrange) equations of motion. On the other hand, for some applications, a simpler on-shell BRST formulation (where the Lautrup-Nakanishi field $B^a$ has been integrated ...


2

Non trivial holonomies have been proposed for quantum computation, see this article http://arxiv.org/pdf/quant-ph/0007110v2.pdf The basic idea is this: Suppose you have a sistem prepared in the ground state of an Hamiltonian $H(\lambda)$, where $\lambda$ is a set of parameters. If you slowly change this parameters the state evolves remaining in the ground ...


6

1) Gauge theory is a theory where we use more than one label to label the same quantum state. 2) Gauge “symmetry” is not a symmetry and can never be broken. This notion of gauge theory is quite unconventional, but true. When two different quantum states $|a\rangle$ and $|b\rangle$ (i.e. $\langle a|b\rangle=0$) have the same properties, we say that there ...


2

OP is pondering if extremization of the action commutes with the minimal coupling (MC) recipe, i.e., if the following diagram would commute: $$ \begin{array}{ccc} \text{Lagrangian density} && \text{Lagrangian density} \cr {\cal L}(\phi(x),\partial\phi(x),A(x),F(x),x) &\stackrel{\text{MC}}{\longrightarrow} & {\cal ...


5

I am not sure of what you mean by "make a difference". The covariant derivative is introduced in the Lagrangian density to add Local Gauge Symmetry to the Action. A given field theory is described by its Action. Hence if the Action does not have a particular symmetry, you cannot introduce the symmetry later. By "making a difference", if you mean a ...


1

There are a lot of rather different aspects being touched on in the question. I'll try to give some indications. But I notice that the relation of this question to actual physics is not very strong, instead the question seems to be more generally after getting a feeling for identity types in homotopy type theory (HoTT). I imagine there are other discussion ...


2

In quantum field theory, when manipulating the path integral, we naively assume the measures (or strictly speaking the product of the measures and integrand) are invariant under the gauge transformations. In a fundamental paper, Fujikawa demonstrated the flaw in this assumption (in certain cases), and how to rigorously compute the analogue of a Jacobian ...



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