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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