Nuclear Physics B (Proc. Suppl.) 34 (1994) 753-755 North-Holland

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PROCEEDINGS SUPPLEMENTS

A New Approach to Numerical Quantum Field Theory Santiago Garcia ~ *

~Department of Physics, Brown University, Providence RI 02912

A new numerical method for solving Lattice Quantum Field Theory is presented. In contrast to Monte-Carlo methods, the Source Galerkin Methodis deterministic and treats fermions and bosons in a more equivalent manner.

1. M o t i v a t i o n

Although Monte-Carlo methods [1] are widely used in Lattice Field Theory, the persistence of difficulties when dealing with fermionic theo- ries (minus sign problem in strongly correlated electron systems, dynamical fermions in Lat- tice QCD) justifies the study of alternative ap- proaches to numerical Quantum Field Theory. In this note, we illustrate with a simple example the Source Galerkin Melhod, which has been de- veloped by the Lattice Theory group at Brown University (G. Guralnik, S.Garcla, J. Lawson, S. Hahn and K. Platt) . Only boson fields will be treated here, the talk of J. Lawson in this Sym- posium will deal with the fermionic case [2].

2. D i f f e r e n t i a l f o r m h l a t i o n

We study a scalar (I ~4 theory in the presence of an external source J(z). d~ and J are defined on a periodic D-dimensional lattice, with Euclidean action

g 4 SE= 1¢2 ~ ( -O+ M2)~y~y + ~Cb~- (1)

where

( - D + M2)x,y - (2D + M 2) 6x,y - Z 6y,x+# (2)

The lattice spacing a has been absorbed in the definition of the sources and parameters, and the

summation is over nearest neighbors. The gen- erating function Z(J) satisfies the the following set of coupled partial differential equations in

*Resea rch s u p p o r t e d in p a r t by NSF G r a n t ASC-9211072 a n d D O E G r a n t D E - F G 0 2 - 9 1 E R 4 0 6 8 8 - Task D

source space

(_n + M2). y OZ 03Z o - S + g - = O (3)

There is one equation for each J=, but note that all equations are related by lattice symmetry op- erations. Our task consists now in (approxi- mately) solving equation (3) with appropriate boundary conditions.

3. T h e G a l e r k i n m e t h o d

Here we discuss briefly the Galerkin method [3]. Consider a linear differential equation

L f = b (4)

L is a differential operator, and b a known func- tion of the N variables x -- ( z l , . . . , X N ) . The solution is sought in a N-dimensional domain 79

The Galerkin method assumes that the exact solution f = f (x ) can be approximated by f~(

K

f~f(x) ---- ~0(x) + Z a j ~j(x) (5) j=l

where ~0 is introduced to satisfy the boundary conditions and the aj's are coefficients to be de- termined. The K basis functions ~j incorporate all our knowledge about symmetries, shape, etc. of f . Substitution of (5) in (4) produces a resid- ual R

K

R(a; x) = L(f*) = L(~o) + Z aj L(~j) (6) j----1

The unknowns aj are then computed by solving the following system of linear algebraic equations

/ vdNxR~k=/vdNxb~bk (7)

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754 S. Garcla/A new approach to numerical quantum field theory

K r

' E aj / _ dNz %5k L~j (s) j = l

: / dN x ~k b - /z) dN x ~gk L~o

where ~ are test functions. Under very gen- eral conditions [3] the Galerkin approximation f ~ converges to f in the mean

l i m / v II/k - Sll = 0 (9)

4. B o u n d a r y C o n d i t i o n s

We must specify boundary conditions for the solution of (3). We normalize the vacuum ampli- tude

Z(J = O) = 1 (10)

and since the differential operator is odd, we re- quire odd derivatives (Green's function) to vanish

O'~ Z J=0 OJ, l . . .OJ~, = 0 , n = l , 3 . . . (11)

Furthermore, we demand Z to be invariant un- der lattice symmetry operations, and to approach smoothly the free field parti t ion function

Zo(J) = e Jx(-n+M~)z,~J~ (12)

when g --+ 0 +. These requirements, that uniquely determine Z, are implicit boundary conditions of all Monte-Carlo (path integral) simulations. Other classes of solutions to the field equations ([4], [5], [6]) may also be of physical interest.

One way of implementing the boundary con- ditions discussed above is to truncate Z in the sources J at some finite order K.

Z(J) = E E a(")P('~)(J) (13) rY n -~ O

where P('~)(J) are lattice invariant polynomials of order n corresponding to a particular symme- try class c~. The coefficients a (n) will be (propor- tional to) the euclidean Green's functions. Note that a Taylor series solution of (3) exists, be- cause the differential operator is not singular at

J = 0, but since the number of polynomials P('~)

grows exponentially with the lattice size, this can only be useful for relatively small lattices. Other more structured approximation schemes will be presented elsewhere.

5. E x a m p l e

As an illustration, we solve (3) for the case of D = 1 and eleven sources J~, x = 1, 2 . . . , 11. We fix M = 1 and g = 0.5 and approximate Z as

a linear combination Z~" 1 of polynomials P('~) up to order n = 4,6,8. There are 6 independent polynomials of order 2, 56 of order 4 ,392 of order 6 and 2052 of order 8. We study the convergence of the different Galerkin estimates for the Green's functions.

After substituting Z~I in (3), a set of inconsis- tent equations for the a (") is obtained, due to the effect of truncation in multiple dimensions. The Galerkin method remedies this unpleasant fact by averaging the residual (error) of the equation. Fortran codes to generate the invariant polyno- mials and to compute and solve numerically the Galerkin equations have been developed [6].

A prescription for test functions that we found convenient, since it provides the necessary num- ber of independent equations for the coefficients

a (n), is the derivative of the basis polynomials re- spect to a source J ,

cgJx (14)

Lattice symmetry makes any choice of x equiva- lent. We set z = 1.

Since we are interested in small J , we choose a hypercube of size 2e centered at the origin as integration domain D, with e very small, and we extrapolate numerically to e = 0.

We show our results for the two point function

G~j = (¢~¢i) - 02Z ' OJiOJj J=0

in table 1. From our simulations of small lat- tices using the symbolic program ALJABR, we

know that the coefficients a~ ~) are rational func- tions of g, M and c, and that a(on)(9, M , Q

a(~)(g, M, 0) + O(e 2) . The limit e ~ 0 can then

,.7. Gareia /A new approach to numerical quantum field theory 755.

Table 1 Results for the two point function Gi,j in a N = l l lattice (anharmonic oscillator). Columns: li - Jl, 4th., 6th. and 8th. order Galerkin approximations, numerical extrapolation (Shanks transformation) and Monte-Carlo data with statistical error.

i - j j 4 j 6 j s S h a n k s M - C

0 0.326538 0.360178 0.347223 0.35082 0.3510 ± 0.0005 1 0.094508 0.117673 0.108121 0.11091 0.1110 ± 0.0004 2 0.026768 0.038823 0.033422 0.03509 0.0351 ± 0.0003 3 0.007454 0.012940 0.010269 0.01114 0.0112 ± 0.0004 4 0.002075 0.004461 0.003197 0.00363 0.0036 ± 0.0003 5 0.000694 0.001922 0.001218 0.00147 0.0011 ± 0.0004

be computed numerically using Richardson ex- trapolation techniques. Calculations were done twice, first with c = 0.3 and then with c = 0.1. The numbers at table 1 are the extrapolation to c = 0. We see that the convergence is oscillatory, and the limit of infinite order in J can be ob- tained by a suitable transformation (Shanks, for instance). The results are in excellent agreement with Monte-Carlo.

The calculations were done on a Cray-2 at NERSC. We let the Monte-Carlo run for two CPU hours, our Galerkin calculations were completed in less than one. This comparison is not very meaningful, since neither our Monte-Carlo nor our Galerkin code were optimized for speed. Be- sides, orders of magnitude respect to Monte-Carlo in complex problems can be gained with more structured approximations that we will present elsewhere. We also gain in quality, since there is no probabilistic noise and the lattice invariance can be easily incorporated.

6. C o n c l u s i o n s

We have discussed here the main ideas of a new method for numerical Quantum Field The- ory, the Source Galerkin Method. This method is applied to solve the discretized functional equa- tions satisfied by Z in source space. We expand Z in suitable lattice symmetric basis functions, and the projection of the equation residual on some test functional space is forced to vanish. Physical boundary conditions for the solution of the differ- ential equation have been given. The method is

very stable, and the results extraordinarily good.

Acknowledgemen t s . I would like to thank the organizers of the Symposium for the stimulat- ing atmosphere we all have enjoyed. G. Guralnik is the source of the ideas exposed in this paper. Jim O'Dell gave us considerable help with AL- JABR, and we had fruitful discussions with G. Kilcup. I am especially grateful to Roberto and Ana Espinosa for their hospitality.

R E F E R E N C E S

1. M. Creutz, L. Jacob and C. Rebbi, Phys. Rev. Lett. 42 (1979) 1390, Claudio Rebbi (ed.), Lattice Gauge Theories and Monte Carlo Simulations, World Scientific, 1983.

2. G. Guralnik and J. Lawson, to be published 3. C.A.J. Fletcher, Computational Galerkin

Methods, Springer-Verlag, 1984. 4. C.M. Bender, F. Cooper and L.M. Simmons,

Phys. Rev D 39, (1989) 2343. 5. E.R. Caianello and G.Scarpetta, Nuovo Ci-

mento A22, (1974) 454. 6. S. Garcia and G. Guralnik, to be published;

S. Garcia, Brown University Ph.D. thesis.