Computational Physics

15.7 Gauge field theories 527

The equations of motion for the fieldAand its conjugate momentumPare then
given by

A ̇n=Pn; (15.170a)

P ̇n=−∂SBoson

∂An

−

∑

lm

φl†

∂[Wlm−^1 (A)]

∂An

φm. (15.170b)

The difficult part is the second equation which involves the derivative of the inverse
ofW(A). The key observation is now that

∂W−^1 (A)

∂An

=W−^1 (A)

∂W(A)

∂An

W−^1 (A), (15.171)

so that we need the vectorηwith

W(A)η=φ. (15.172)

This can be found using a suitable sparse matrix algorithm. Using thisη-field, the
equation of motion forPsimply reads:

P ̇n=−∂SBoson

∂An

−η†

∂(M†M)

∂An

η. (15.173)

Summarising, a molecular dynamics update consists of the following steps:

ROUTINE MDStep

Generate a Gaussian random configurationξ;

Calculateφ=M(A)ξ;

Calculateηfrom(M†M)η=φ;

Update the boson fieldAand its conjugate momentum fieldPusing

P(t+h/ 2 )=P(t−h/ 2 )−h

[

∂SBoson

∂An

+η

∂(MTM)

∂An

η

]

and

A(t+h)=A(t)+hP(t+h/ 2 ).

END MDStep

We see that in both the Langevin and the hybrid method, the most time-consuming
step is the calculation of a (sparse) matrix equation at each field update step (in the
above algorithm this is the step in the third line).

15.7.5 Non-abelian gauge fields; quantum chromodynamics

QED is the theory for charged fermions interacting through photons, which are
described by a real-valued vector gauge fieldAμ. Weak and strong interactions
are described by similar but more complicated theories. A difference between