Suppose we know a classical solution for gauge Boson fields $A_{\mu,c}$ and Fermion fields $\psi_c$ and now we want to consider ist Quantum fluctuations. These fluctuations arise from loop corrections, where virtual off-Shell particles are present for a short time.

We can assume each virtual pair of particles/antiparticles as a Wilson loop over a closed curve $\gamma$

$W_\gamma = P exp(i \int_\gamma dx^\mu A_\mu)$.

Moreover, we introduce another separate Quantum field $B_{\mu \nu}$, which Plays the role as the Yang-Mills field strength. From now on I will use differential forms instead of coordinate representation. Let $\Sigma$ be a closed Surface. Then we can introduce the variable

$X_{\Sigma,C_{x_0}} = exp(i \int_\Sigma Hol(x_0, p \in \Sigma)B(p \in \Sigma)Hol^*(p \in \Sigma,x_0))$

with a holonomy $Hol(x_0,p)$ from the point $x_0$ to a point on $\Sigma$ such that when continuing the *-holonomy, the colsed loop $C_{x_0}$ with base point $x_0$ is formed.

Now I use the almost topological Lagrangian

$L = \int (B \wedge (dA + A \wedge A) + J_A \wedge *A + J_B \wedge *B)$ (BF Theory with some Sources $J_A,J_B$ depending e.g. on the classical Dynamics; in pure vacuum These are zero). The dynamical Hamiltonian is assumed as being Zero for These vacuum excitations.

and can compute observable expectation values with the path integral. Now $O$ is an arbitrary product of observables given above. Using the field variable shift invariance in path integral, we arrive at the Quantum equations of Motion ($<\dots>$ denotes averaging with weight $Oe^{iS}$):

$d<A> + <A \wedge A> = - J_B -i <\frac{1}{O}\frac{\delta O}{\delta B}>$ (1),

$d<B> + <[A, \wedge B]_-> = - J_A -i <\frac{1}{O}\frac{\delta O}{\delta A}>$ (2).

Now we notice that (2) is the generalized Maxwell equation with classical particle current $J_A$ and equation (1) the Expression for the field strength (classical values are $J_B$). The observables $O$ are arbitrary, so These equations describe a Quantum System with fixed physical properties depending on the choice of $O$, for example $<\frac{1}{O}\frac{\delta O}{\delta A}> \propto \dot \gamma$ at the Point, where the Wilson loop has Tangent Vector $\dot \gamma$.

But a lot more important is computing the probability Amplitude that Quantum fluctuations given by the form of $W_\gamma,X_{\Sigma,C_{x_0}}$ will occur. That means, we want to compute $A = <1>/<O^{-1}>$.

The Question: Is it an Alternate way to compute Transition amplitudes with respecting also the intermediate states (popping in and out of particles) to use above topological Quantum field Theory? Would it be more practical in comparison with typical computation methods like Lattice models in some cases?

  • $\begingroup$ That theory is not topological once you properly impose the constraint $B = \star F$, since the Hodge star contains a factor of the volume form. $\endgroup$ – Ryan Thorngren Aug 24 '18 at 17:07
  • $\begingroup$ I have assumed that the term $B \wedge *B$ can be set to Zero for the vacuum fluctuation observables. $\endgroup$ – kryomaxim Aug 24 '18 at 18:47
  • $\begingroup$ Could you clarify your question? Is your question how to compute correlation functions of observations in BF theory? And perhaps clean up your notation? (Some of the equations are inconsistent as written. Also, for simplcity, use a 3-form $J_A$ and a 2-form $J_B$. Then the hodge star drops out.) $\endgroup$ – user1504 May 17 at 21:16

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