Sun Aug 18 16:53:00 2002 run on Windows
COMMENT
test file for the PHYSOP package;
% load_package physop; % Load a compiled version of the physop package.
% showtime;
linelength(72)$
% Example 1: Quantum Mechanics of a Dirac particle in an external
% electromagnetic field
VECOP P,A,K;
SCALOP M;
NONCOM P,A;
PHYSINDEX J,L;
oporder M,K,A,P;
% we have to set off allfac here since otherwise there appear
% spurious negative powers in the printed output
off allfac;
FOR ALL J,L LET COMM(P(J),A(L))=K(J)*A(L);
H:= COMMUTE(P**2/(2*M),E/(4*M**2)*(P DOT A));
-1 -1 -1
h := (2*e*(m )*(m )*(m )*k(idx1)*a(idx2)*p(idx1)*p(idx2)
-1 -1 -1
+ e*(m )*(m )*(m )*k(idx1)*k(idx1)*a(idx2)*p(idx2)
-1 -1 -1
+ e*(m )*(m )*(m )*k(idx1)*k(idx1)*k(idx2)*a(idx2)
-1 -1 -1
+ 2*e*(m )*(m )*(m )*k(idx1)*k(idx2)*a(idx2)*p(idx1))/8
% showtime;
%assign the corresponding value to the adjoint of H
H!+ := adj H;
+ + + + + -1 -1
(h ) := (e*(a(idx2) )*(k(idx1) )*(k(idx1) )*(k(idx2) )*(m!+ )*(m!+ )
-1 + + + +
*(m!+ ) + 2*e*(p(idx1) )*(a(idx2) )*(k(idx1) )*(k(idx2) )
-1 -1 -1 + + +
*(m!+ )*(m!+ )*(m!+ ) + 2*e*(p(idx1) )*(p(idx2) )*(a(idx2) )
+ -1 -1 -1 + +
*(k(idx1) )*(m!+ )*(m!+ )*(m!+ ) + e*(p(idx2) )*(a(idx2) )
+ + -1 -1 -1
*(k(idx1) )*(k(idx1) )*(m!+ )*(m!+ )*(m!+ ))/8
% showtime;
% note the ordering of operators in the result!
% enhance the readability of the output
on allfac;
ON CONTRACT;
H;
3
(e*m!-1
2 2
*(2*a dot p*k dot p + 2*k dot a*k dot p + k *a dot p + k *k dot a))/8
% showtime;
% Example 2: Virasoro Algebra from Conformal Field Theory
operator del;
% this is just a definition of a delta function
for all n such that numberp n let del(n) =
if n=0 then 1
else 0;
scalop l;
noncom l,l;
state bra,ket;
% commutation relation of the operator l;
for all n,m let comm(l(n),l(m)) =
(m-n)*l(n+m)+c/12*(m**3-m)*del(n+m)*unit;
%modified 1.1
for all n let l!+(n) = l(-n);
% relation for the states
for all h let bra!+(h) = ket(h);
for all p,q let bra(q) | ket(p) = del(p-q);
for all r,h such that r < 0 or (r <2 and h=0) let
l(r) | ket(h) = 0;
for all r,h such that r > 0 or (r > -2 and h = 0) let
bra(h) | l(r) = 0;
% define a procedure to calculate V.E.V.
procedure Vak(X);
bra(0) | X | ket(0);
vak
% and now some calculations;
MA:= adj(l(3)*l(5))*l(3)*l(5);
ma := 2*l(8)*l(-3)*l(-5) + 4*l(8)*l(-8) + l(5)*l(3)*l(-3)*l(-5)
+ 2*l(5)*l(3)*l(-8) + 6*l(5)*l(0)*l(-5) + 8*l(5)*l(-2)*l(-3)
+ 60*l(5)*l(-5) + 8*l(3)*l(2)*l(-5) + 10*l(3)*l(0)*l(-3)
2
+ 112*l(3)*l(-3) + 64*l(2)*l(-2) + 60*l(0) + 556*l(0)
2
+ 20*c *unit + 2*c*l(5)*l(-5) + 10*c*l(3)*l(-3) + 80*c*l(0)
+ 332*c*unit
%modified 1.1
% showtime;
% here is the VEV of m
vak(Ma);
4*c*(5*c + 83)
% showtime;
% and now calculate another matrix element
matel := bra(1) | ma | ket(1);
*************** WARNING: ***************
Evaluation incomplete due to missing elementary relations
matel := bra(1) | (l(0) | 556*ket(1)) + bra(1) | (l(0) | 80*c*ket(1))
2
+ bra(1) | (l(0)*l(0) | 60*ket(1)) + 20*c + 332*c
%modified 1.1
% showtime;
% this evaluation is incomplete so supply the missing relation
for all h let l(0) | ket(h) = h*ket(h);
% and reevaluate matel
matel := matel;
2
matel := 4*(5*c + 103*c + 154)
% showtime;
% Example 4: some manipulations with gamma matrices to demonstrate
% the use of commutators and anticommutators
off allfac;
vecop gamma,q;
tensop sigma(2);
antisymmetric sigma;
noncom gamma,gamma;
noncom sigma,gamma;
physindex mu,nu;
operator delta;
for all mu,nu let anticomm(gamma(mu),gamma(nu))=2*delta(mu,nu)*unit,
comm(gamma(mu),gamma(nu))=2*I*sigma(mu,nu);
oporder p,q,gamma,sigma;
off allfac;
on anticom;
(gamma dot p)*(gamma dot q);
p(idx4)*q(idx5)*gamma(idx4)*gamma(idx5)
% showtime;
off anticom;
(gamma dot p)*(gamma dot q);
p(idx6)*q(idx7)*gamma(idx6)*gamma(idx7)
% showtime;
commute((gamma dot p),(gamma dot q));
2*i*p(idx8)*q(idx9)*sigma(idx8,idx9)
% showtime;
anticommute((gamma dot p),(gamma dot q));
2*p(idx10)*q(idx11)*gamma(idx10)*gamma(idx11)
- 2*i*p(idx10)*q(idx11)*sigma(idx10,idx11)
on anticom;
anticommute((gamma dot p),(gamma dot q));
2*delta(idx13,idx12)*p(idx12)*q(idx13)
% showtime;
end;
Time for test: 1072 ms