Lattice Gauge Theory and the Maxwell-Klein-Gordon equations

Lattice Gauge Theory and the Maxwell-Klein-Gordon equations Tore G. Halvorsen Centre of Mathematics for Applications, UiO 19. February 2008

Abstract In this talk I will present a discretization of the Maxwell-Klein-Gordon equations, motivated by Lattice Gauge Theory, which preserves the local gauge invariance. Due to this symmetry, the electric charge of the system is conserved, which is not the case when using a standard leap-frog discretization. Lastly, some numerical results will be presented.

Outline

The space-time domain under consideration is Ω = R 3 R, (1) with coordinates x = (x,t) Ω, and Minkowski metric η µν =diag(1,1,1, 1) The unknowns are A complex scalar field x φ(x) C A real gauge potential x A µ (x) R, µ = 0,x,y,z.

We also introduce Field strenght F µν (x) = µ A ν (x) ν A µ (x), the covariant derivative µ := x µ (2) the electric- and magnetic fields D µ = µ ia µ, (3) E i (x) = F i0 (x) B i (x) = 1 2 ε ijkf jk (x). (4)

The MKG equations are the Euler-Lagrange equations of the following action S[A,φ] = dx( 1 4 F µνf µν + (D µ φ)(d µ φ) + m 2 φ 2 ), (5) Ω and the Bianchi identity for the field strength ( Dµ D µ m 2) φ = 0 (6) ν F µν + J µ = 0, µ λ F µν + µ F νλ + ν F λµ = 0 (7) where J µ = i(φ D µ φ φd µ φ ).

Equation 7 can further be divided into evolution equations E t = curlb + J and constraint equations B t = curle, (8) dive + J 0 = 0, divb = 0. (9) These are the Maxwell s equations with source.

Invariance under local U(1) transformations φ(x) φ(x)e iλ(x) A µ (x) A µ (x) + µ λ(x), λ(x) R (10) The gauge symmetry can be viewed as an analogue to the equivalence principle in General Relativity. Noether s theorem implies that the constraint, is conserved 0 (dive + J 0 ) = 0. (11)

In the spirit of LGT, one can formulate a discrete gauge invariant theory for the MKG case. This prosess can be divided into three steps:

In the spirit of LGT, one can formulate a discrete gauge invariant theory for the MKG case. This prosess can be divided into three steps: Introduce a space-time lattice, L, with lattice points x = (x,t) and lattice spacing a µ

In the spirit of LGT, one can formulate a discrete gauge invariant theory for the MKG case. This prosess can be divided into three steps: Introduce a space-time lattice, L, with lattice points x = (x,t) and lattice spacing a µ Discretize the Klein-Gordon part of the action on the lattice as S[φ] = ((D µ φ)(d µ φ) + m 2 φ 2 ) h(a) L (D µ φ)(d µ φ) + m 2 φ 2 := S KG [φ,a] D µ φ(x) 1 ( ) φ(x + a µ ) e iaµaµ(x+ 1 2 aµ) φ(x), h(a) = a t ax 3 a i µ

e iaµaµ(x+ 1 2 aµ) is called a link variable and is introduced to compensate for the difference in phase transformations from one point to the next. If e iaµaµ(x+ 1 2 aµ) 1 + ia µ A µ (x + 1 2 a µ) derivatives are approximated by forward Euler and the coupling between A µ and φ is minimal.

e iaµaµ(x+ 1 2 aµ) is called a link variable and is introduced to compensate for the difference in phase transformations from one point to the next. If e iaµaµ(x+ 1 2 aµ) 1 + ia µ A µ (x + 1 2 a µ) derivatives are approximated by forward Euler and the coupling between A µ and φ is minimal. Add a 2. order Yee-type action for the Maxwell part S Yee [A] = h(a) ( ) 1 4 F µνf µν, (12) L

F µν (x) = 1 a µ (A ν (x + 1 2 a ν + a µ ) A ν (x + 1 2 a ν)) 1 a ν (A µ (x + 1 2 a µ + a ν ) A µ (x + 1 2 a µ)) The discrete action we are using is hence S MKG [φ,a] = S KG [φ,a] + S Yee [A], (13) and we see that it is locally gauge invariant in the sence φ(x) φ(x)e iλ(x) A µ (x + 1 2 a µ) A µ (x + 1 2 a µ) + 1 a µ (λ(x + a µ ) λ(x)). (14)

This action gives the following discrete E-L equations φ : 1 a 2 t 1 a 2 i ( ) φ(x + a t )e iata 0(x+ 1 2 at) + φ(x a t )e iata 0(x 1 2 at) 2φ(x) ( ) φ(x + a i )e ia ia i (x+ 1 2 ai) + φ(x a i )e ia ia i (x 1 2 ai) 2φ(x) + m 2 φ(x) = 0, (15)

A 0 : i 1 i F 0i (x)) + 1 ( i φ(x + a t )φ (x)e iata 0(x+ 1 2 at) a i a t φ(x)φ (x + a t )e iata 0(x+ 1 2 at)) = 0 (16) This equation represents the constraint equation dive + J 0 = 0 in the discrete case. Again it follows from Noether s theorem that this quantity is conserved

A i : 1 t F 0i (x) 1 j F ji (x)+ a t a j j ( 1 i φ(x + a i )φ (x)e ia ia i (x+ 1 2 ai) a i φ(x)φ (x + a i )e ia ia i (x+ 1 2 a i) ) = 0, (17) These equations together with the discrete Bianchi identity for the field strenght comprise the MKG equations in the discrete case.

Above described scheme, Leap-Frog scheme Implemented on [0,1] 3 [0,1], N i = 30 with Periodic Boundary Conditions, N t = 100 Initialized as plane waves dive + J 0 L 2 and dive L 2

Figure: dive + J 0 L 2 and dive L 2 as a function of time 0.25 div E L 2 vs. div E + J 0 L 2 0.25 div E L 2 vs. div E + J 0 L 2 div E+J 0 L 2 0.2 div E L 2 0.2 div E + J 0 L 2 div E L 2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0.5 1.0 t 0 0 0.5 1.0 t (a) Standard scheme (b) LGT scheme

Conclusion Gauge invariance conserved charge Discretization through link variables. General procedure, applicable to other equations with gauge symmetry

Conclusion Gauge invariance conserved charge Discretization through link variables. General procedure, applicable to other equations with gauge symmetry Convergence Formulation on a general Riemannian manifold Expand the scheme to the more general Yang-Mills-Higgs equations

References Heinz J. Rothe; Lattice Gauge Theories: An Introduction; World Scientific Lecture Notes in Physics - Vol.74; 2005 M. Creutz; Quarks, gluons and lattices; Cambridge Monographs On Mathematical Physics; 1983 Kenneth G. Wilson, Confinement of quarks, Phys. Rev. D 10(8) 1974