REDUCE 3.4.1, 15-Jul-92 ...
1:
(PHYSOP)
COMMENT
test file for the PHYSOP package;
% load_package physop; % Load a compiled version of the physop package.
showtime;
Time: 0 ms
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 := (E*(M )*(M )*(M )*K(IDX1)*K(IDX1)*K(IDX2)*A(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(IDX2)*A(IDX2)*P(IDX1)
-1 -1 -1
+ E*(M )*(M )*(M )*K(IDX1)*A(IDX2)*P(IDX1)*P(IDX2)
-1 -1 -1
+ E*(M )*(M )*(M )*K(IDX1)*K(IDX2)*A(IDX2)*P(IDX1)
-1 -1 -1
+ E*(M )*(M )*(M )*K(IDX1)*A(IDX2)*P(IDX1)*P(IDX2))
/8
showtime;
Time: 272 ms
%assign the corresponding value to the adjoint of H
H!+ := adj H;
+ + + + +
(H ) := (2*E*(P(IDX1) )*(P(IDX2) )*(A(IDX2) )*(K(IDX1) )
-1 -1 -1 + +
*(M!+ )*(M!+ )*(M!+ ) + 2*E*(P(IDX1) )*(A(IDX2) )
+ + -1 -1 -1
*(K(IDX1) )*(K(IDX2) )*(M!+ )*(M!+ )*(M!+ ) + E
+ + + + -1
*(P(IDX2) )*(A(IDX2) )*(K(IDX1) )*(K(IDX1) )*(M!+ )
-1 -1 + +
*(M!+ )*(M!+ ) + E*(A(IDX2) )*(K(IDX1) )
+ + -1 -1 -1
*(K(IDX1) )*(K(IDX2) )*(M!+ )*(M!+ )*(M!+ ))/8
showtime;
Time: 272 ms
% note the ordering of operators in the result!
% enhance the readability of the output
on allfac;
ON CONTRACT;
H;
3 2 2
(E*M!-1 *(K *K DOT A + K *A DOT P + 2*A DOT P*K DOT P
+ 2*K DOT A*K DOT P))/8
showtime;
Time: 102 ms
% 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);
2
MA := 20*C *UNIT + 332*C*UNIT + 2*C*L(5)*L(-5)
+ 10*C*L(3)*L(-3) + 80*C*L(0) + 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)
%modified 1.1
showtime;
Time: 357 ms
% here is the VEV of m
vak(Ma);
4*C*(5*C + 83)
showtime;
Time: 119 ms
% and now calculate another matrix element
matel := bra(1) | ma | ket(1);
*************** WARNING: ***************
Evaluation incomplete due to missing elementary relations
2
MATEL := 20*C + 332*C + BRA(1) | (L(0) | 556*KET(1))
+ BRA(1) | (L(0) | 80*C*KET(1))
+ BRA(1) | (L(0)*L(0) | 60*KET(1))
%modified 1.1
showtime;
Time: 102 ms
% 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;
Time: 34 ms
% 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;
Time: 51 ms
off anticom;
(gamma dot p)*(gamma dot q);
P(IDX6)*Q(IDX7)*GAMMA(IDX6)*GAMMA(IDX7)
showtime;
Time: 17 ms
commute((gamma dot p),(gamma dot q));
2*I*P(IDX8)*Q(IDX9)*SIGMA(IDX8,IDX9)
showtime;
Time: 17 ms
anticommute((gamma dot p),(gamma dot q));
- 2*I*P(IDX10)*Q(IDX11)*SIGMA(IDX10,IDX11)
+ 2*P(IDX10)*Q(IDX11)*GAMMA(IDX10)*GAMMA(IDX11)
on anticom;
anticommute((gamma dot p),(gamma dot q));
2*DELTA(IDX13,IDX12)*P(IDX12)*Q(IDX13)
showtime;
Time: 85 ms
end;
Time: 0 ms
Quitting