Question

Is Yang-Mills theory as a topological quantum field theory a good alternative?

Answer 1

Suppose we know a classical solution for gauge Boson fields Aμ,cAμ,c and Fermion fields ψcψc 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 γγ

Wγ=Pexp(i∫γdxμAμ)Wγ=Pexp(i∫γdxμAμ).

Moreover, we introduce another separate Quantum field BμνBμν, which Plays the role as the Yang-Mills field strength. From now on I will use differential forms instead of coordinate representation. Let ΣΣbe a closed Surface. Then we can introduce the variable

XΣ,Cx0=exp(i∫ΣHol(x0,p∈Σ)B(p∈Σ)Hol∗(p∈Σ,x0))XΣ,Cx0=exp(i∫ΣHol(x0,p∈Σ)B(p∈Σ)Hol∗(p∈Σ,x0))

with a holonomy Hol(x0,p)Hol(x0,p) from the point x0x0 to a point on ΣΣ such that when continuing the *-holonomy, the colsed loop Cx0Cx0 with base point x0x0is formed.

Now I use the almost topological Lagrangian

L=∫(B∧(dA+A∧A)+JA∧∗A+JB∧∗B)L=∫(B∧(dA+A∧A)+JA∧∗A+JB∧∗B)(BF Theory with some Sources JA,JBJA,JB 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 OO 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 (<⋯><⋯>denotes averaging with weight OeiSOeiS):

d<A>+<A∧A>=−JB−i<1OδOδB>d<A>+<A∧A>=−JB−i<1OδOδB>(1),

d<B>+<[A,∧B]−>=−JA−i<1OδOδA>d<B>+<[A,∧B]−>=−JA−i<1OδOδA> (2).

Now we notice that (2) is the generalized Maxwell equation with classical particle current JAJA and equation (1) the Expression for the field strength (classical values are JBJB). The observables OO are arbitrary, so These equations describe a Quantum System with fixed physical properties depending on the choice of OO, for example <1OδOδA>∝γ˙<1OδOδA>∝γ˙ at the Point, where the Wilson loop has Tangent Vector γ˙γ˙.

But a lot more important is computing the probability Amplitude that Quantum fluctuations given by the form of Wγ,XΣ,Cx0Wγ,XΣ,Cx0 will occur. That means, we want to compute A=<1>/<O−1>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

Answer 2

That theory is not topological once you properly impose the constraint B=⋆FB, since the Hodge star contains a factor of the volume form.