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- Tue Feb 10 12:26:52 2004 run on Linux
- 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: 30 ms
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