module hephys; % Support for high energy physics calculations.
% Author: Anthony C. Hearn.
% Generalizations for n dimensional vector and gamma algebra by
% Gastmans, Van Proeyen and Verbaeten, University of Leuven, Belgium.
% Copyright (c) 1991 The RAND Corporation. All rights reserved.
create!-package('(hephys),'(appl));
fluid '(!*nospurp !*sub2 ncmp!* ndims!*);
global '(defindices!* indices!* mul!* ndim!*);
defindices!* := nil; % Deferred indices in N dim calculations.
indices!* := nil; % List of indices in High Energy Physics
% tensor expressions.
ndim!* := 4; % Number of dimensions in gamma algebra.
% *********************** SOME DECLARATIONS *************************
deflist ('((cons simpdot)),'simpfn);
flag('(cons),'symmetric); % Since CONS is used in algebraic mode
% only for High Energy Physics expressions.
put('vector,'stat,'rlis);
% put('vector,'formfn,'formvector);
%symbolic procedure formvector(u,vars,mode);
% if mode eq 'algebraic
% then list('vector1,'list . formlis(cdr u,vars,'algebraic))
% else u;
symbolic procedure vector u; vector1 u;
symbolic procedure vector1 u;
for each x in u do
begin scalar y;
if not idp x or (y := gettype x) and y neq 'hvector
then typerr(if y then {y,x} else x,"high energy vector")
else put(x,'rtype,'hvector)
end;
put('hvector,'fn,'vecfn);
put('hvector,'evfn,'veval);
put('g,'simpfn,'simpgamma);
noncom g;
symbolic procedure index u;
begin vector1 u; rmsubs(); indices!* := union(indices!*,u) end;
symbolic procedure remind u;
begin indices!* := setdiff(indices!*,u) end;
symbolic procedure mass u;
if null car u then rerror(hephys,1,"No arguments to MASS")
else <<for each x in u do put(cadr x,'rtype,'hvector);
for each x in u do put(cadr x,'mass,caddr x)>>;
symbolic procedure getmas u;
(if x then x else rerror(hephys,2,list(u,"has no mass")))
where x=get(u,'mass);
symbolic procedure vecdim u;
begin ndim!* := car u end;
symbolic procedure mshell u;
begin scalar x,z;
a: if null u then return let0 z;
x := getmas car u;
z := list('equal,list('cons,car u,car u),list('expt,x,2)) . z;
u := cdr u;
go to a
end;
symbolic procedure nospur u; <<!*nospurp := t; flag(u,'nospur)>>;
symbolic procedure spur u;
<<rmsubs(); remflag(u,'nospur); remflag(u,'reduce)>>;
rlistat '(vecdim index mass mshell remind nospur spur vector);
% ******** FUNCTIONS FOR SIMPLIFYING HIGH ENERGY EXPRESSIONS *********
symbolic procedure veval(u,v);
begin scalar z;
for each x in nssimp(u,'hvector) do
<<if null cdr x then rerror(hephys,3,"Missing vector")
else if cddr x
then msgpri("Redundant vector in",cdr x,nil,nil,t);
z := aconc!*(z,retimes(prepsq car x . cdr x))>>;
return replus z
end;
symbolic procedure vmult u;
begin scalar z;
z := list list(1 . 1);
a: if null u then return z;
z := vmult1(nssimp(car u,'hvector),z);
if null z then return;
u := cdr u;
go to a
end;
symbolic procedure vmult1(u,v);
begin scalar z;
if null v then return;
a: if null u then return z
else if cddar u
then msgpri("Redundant vector in",cdar u,nil,nil,t);
z := nconc!*(z,for each j in v collect
multsq(car j,caar u) . append(cdr j,cdar u));
u := cdr u;
go to a
end;
symbolic procedure simpdot u;
mkvarg(u,function dotord);
symbolic procedure dotord u;
<<if xnp(u,indices!*) and not ('isimpq memq mul!*)
then mul!* := aconc!*(mul!*,'isimpq) else nil;
if 'a memq u
then rerror(hephys,4,
"A represents only gamma5 in vector expressions")
else mksq('cons . ord2(car u,carx(cdr u,'dot)),1)>>;
symbolic procedure mkvarg(u,v);
begin scalar z;
u := vmult u;
z := nil ./ 1;
a: if null u then return z;
z := addsq(multsq(apply1(v,cdar u),caar u),z);
u := cdr u;
go to a
end;
symbolic procedure simpgamma u;
if null u or null cdr u
then rerror(hephys,5,"Missing arguments for G operator")
else begin scalar z;
if not ('isimpq memq mul!*) then mul!*:= aconc!*(mul!*,'isimpq);
ncmp!* := t;
z := nil ./ 1;
for each j in vmult cdr u do
z := addsq(multsq(mksq('g . car u . cdr j,1),car j),z);
return z
end;
symbolic procedure simpeps u;
mkvarg(u,function epsord);
symbolic procedure epsord u;
if repeats u then nil ./ 1 else mkepsq u;
symbolic procedure mkepsk u;
% U is of the form (v1 v2 v3 v4).
% Value is <sign flag> . <kernel for EPS(v1,v2,v3,v4)>.
begin scalar x;
if xnp(u,indices!*) and not('isimpq memq mul!*)
then mul!* := aconc!*(mul!*,'isimpq);
x := ordn u;
u := permp(x,u);
return u . ('eps . x)
end;
symbolic procedure mkepsq u;
(lambda x; (lambda y; if null car x then negsq y else y)
mksq(cdr x,1))
mkepsk u;
% ** FUNCTIONS FOR SIMPLIFYING VECTOR AND GAMMA MATRIX EXPRESSIONS **
symbolic smacro procedure mkg(u,l);
% Value is the standard form for G(L,U).
mksf('g . l . u);
symbolic smacro procedure mka l;
% Value is the standard form for G(L,A).
mksf list('g,l,'a);
symbolic smacro procedure mkgamf(u,l);
mksf('g . (l . u));
symbolic procedure mkg1(u,l);
if not flagp(l,'nospur) then mkg(u,l) else mkgamf(u,l);
symbolic smacro procedure mkpf(u,v);
multpf(u,v);
symbolic procedure mkf(u,v);
multf(u,v);
symbolic procedure multd!*(u,v);
if u=1 then v else multd(u,v); % onep
symbolic smacro procedure addfs(u,v);
addf(u,v);
symbolic smacro procedure multfs(u,v);
% U and V are pseudo standard forms.
% Value is pseudo standard form for U*V.
multf(u,v);
put('rcons,'cleanupfn,'isimpa);
symbolic procedure isimpa(u,v);
if eqcar(u,'list) then u else !*q2a1(isimpq simp u,v);
symbolic procedure isimpq u;
begin scalar ndims!*;
ndims!* := simp ndim!*;
if denr ndims!* neq 1
then <<!*sub2 := t;
ndims!* := multpf(mksp(list('recip,denr ndims!*),1),
numr ndims!*)>>
else ndims!* := numr ndims!*;
a: u := isimp1(numr u,indices!*,nil,nil,nil) ./ denr u;
if defindices!*
then <<indices!* := union(defindices!*,indices!*);
defindices!* := nil;
go to a>>
else if null !*sub2 then return u
else return resimp u
end;
symbolic procedure isimp1(u,i,v,w,x);
if null u then nil
else if domainp u
then if x then multd(u,spur0(car x,i,v,w,cdr x))
else if v
then rerror(hephys,6,"Unmatched index" . mapovercar v)
else if w then multfs(emult w,isimp1(u,i,v,nil,x))
else u
else addfs(isimp2(car u,i,v,w,x),isimp1(cdr u,i,v,w,x));
symbolic procedure isimp2(u,i,v,w,x);
begin scalar z;
if atom (z := caar u) then go to a
else if car z eq 'cons and xnp(cdr z,i)
then return dotsum(u,i,v,w,x)
else if car z eq 'g
then go to b
else if car z eq 'eps then return esum(u,i,v,w,x);
a: return mkpf(car u,isimp1(cdr u,i,v,w,x));
b: z := gadd(appn(cddr z,cdar u),x,cadr z);
return isimp1(multd!*(nb car z,cdr u),i,v,w,cdr z)
end;
symbolic procedure nb u;
if u then 1 else -1;
symbolic smacro procedure mkdot(u,v);
% Returns a standard form for U . V.
mksf('cons . ord2(u,v));
symbolic procedure dotsum(u,i,v,w,x);
begin scalar i1,n,u1,u2,v1,y,z,z1;
n := cdar u;
if not (car (u1 := cdaar u) member i) then u1 := reverse u1;
u2 := cadr u1;
u1 := car u1;
v1 := cdr u;
if n=2 then go to h
else if n neq 1 then typerr(n,"index power");
a: if u1 member i then go to a1
else if null (z := mkdot(u1,u2)) then return nil
else return mkf(z,isimp1(v1,i1,v,w,x));
a1: i1 := delete(u1,i);
if u1 eq u2 then return multf(ndims!*,isimp1(v1,i1,v,w,x))
else if not (z := bassoc(u1,v)) then go to c
else if u2 member i then go to d;
if u1 eq car z then u1 := cdr z else u1 := car z;
go to e;
c: if z := memlis(u1,x)
then return isimp1(v1,
i1,
v,
w,
subst(u2,u1,z) . delete(z,x))
else if z := memlis(u1,w)
then return esum((('eps . subst(u2,u1,z)) . 1) . v1,
i1,
v,
delete(z,w),
x)
else if u2 member i and null y then go to g;
return isimp1(v1,i,(u1 . u2) . v,w,x);
d: z1 := u1;
u1 := u2;
if z1 eq car z then u2 := cdr z else u2 := car z;
e: i := i1;
v := delete(z,v);
go to a;
g: y := t;
z := u1;
u1 := u2;
u2 := z;
go to a1;
h: if u1 eq u2 then rerror(hephys,7,
"2 invalid as repeated index power");
i := i1 := delete(u1,i);
u1 := u2;
go to a
end;
symbolic procedure mksf u;
% U is a non-unique kernel.
% Value is a (possibly substituted) standard form for U.
begin scalar x;
x := mksq(u,1);
if denr x=1 then return numr x;
!*sub2 := t;
return !*p2f mksp(u,1)
end;
% ********* FUNCTIONS FOR SIMPLIFYING DIRAC GAMMA MATRICES **********
symbolic procedure gadd(u,v,l);
begin scalar w,x; integer n;
n := 0; % Number of gamma5 interchanges.
if not (x := atsoc(l,v)) then go to a;
v := delete(x,v);
w := cddr x; % List being built.
x := cadr x; % True if gamma5 remains.
a: if null u then return (evenp n . (l . x . w) . v)
else if car u eq 'a then go to c
else w := car u . w;
b: u := cdr u;
go to a;
c: if ndims!* neq 4
then rerror(hephys,8,"Gamma5 not allowed unless vecdim is 4");
x := not x;
n := length w + n;
go to b
end;
% ***** FUNCTIONS FOR COMPUTING TRACES OF DIRAC GAMMA MATRICES *******
symbolic procedure spur0(u,i,v1,v2,v3);
begin scalar l,w,i1,kahp,n,z;
l := car u;
n := 1;
z := cadr u;
u := reverse cddr u;
if z then u := 'a . u; % Gamma5 remains.
if null u then go to end1
else if null flagp(l,'nospur)
then if car u eq 'a and (length u<5 or hevenp u)
or not(car u eq 'a) and not hevenp u
then return nil
else if null i then <<w := reverse u; go to end1>>;
a:
if null u then go to end1
else if car u member i
then if car u member cdr u
then <<if car u eq cadr u
then <<i := delete(car u,i);
u := cddr u;
n := multf(n,ndims!*);
go to a>>;
kahp := t;
i1 := car u . i1;
go to a1>>
else if car u member i1 then go to a1
else if z := bassoc(car u,v1)
then <<v1 := delete(z,v1);
if w then i := delete(car w,i);
u := other(car u,z) . cdr u;
go to a>>
else if z := memlis(car u,v2)
then return if flagp(l,'nospur)
and null v1
and null v3
and null cdr v2
then mkf(mkgamf(append(reverse w,u),l),
multfs(n,mkepsf z))
else multd!*(n,
isimp1(spur0(
l . (nil . append(reverse u,w)),nil,nil,delete(z,v2),v3),
i,v1,list z,nil))
else if z := memlis(car u,v3)
then if ndims!*=4
then return spur0i(u,delete(car u,i),v1,v2,
delete(z,v3),l,n,w,z)
else <<indices!* := delete(car u,indices!*);
i := delete(car u,i);
if not(car u memq defindices!*)
then defindices!* :=
car u . defindices!*;
go to a1>>
else rerror(hephys,9,list("Unmatched index",car u));
a1:
w := car u . w;
u := cdr u;
go to a;
end1:
if kahp
then if ndims!*=4
then <<z := multfs(n,kahane(reverse w,i1,l));
return isimp1(z,setdiff(i,i1),v1,v2,v3)>>
else z := spurdim(w,i,l,nil,1)
else z := spurr(w,l,nil,1);
return if null z then nil
else if get('eps,'klist) and not flagp(l,'nospur)
then isimp1(multfs(n,z),i,v1,v2,v3)
else multfs(z,isimp1(n,i,v1,v2,v3))
end;
symbolic procedure spur0i(u,i,v1,v2,v3,l,n,w,z);
begin scalar kahp,i1;
if flagp(l,'nospur) and flagp(car z,'nospur)
then rerror(hephys,10,
"NOSPUR on more than one line not implemented")
else if flagp(car z,'nospur) then kahp := car z;
z := cdr z;
i1 := car z;
z := reverse cdr z;
if i1 then z := 'a . z;
i1 := nil;
<<while null (car u eq car z) do
<<i1 := car z . i1; z := cdr z>>;
z := cdr z;
u := cdr u;
if flagp(l,'nospur)
then <<w := w . (u . (i1 . z));
i1 := car w;
z := cadr w;
u := caddr w;
w := cdddr w>>;
w := reverse w;
if null ((null u or not eqcar(w,'a)) and (u := append(u,w)))
then <<if not hevenp u then n := - n;
u := 'a . append(u,cdr w)>>;
if kahp then l := kahp;
z :=
mkf(mkg(reverse i1,l),
multf(brace(u,l,i),multfs(n,mkg1(z,l))));
z := isimp1(z,i,v1,v2,v3);
if null z or (z := quotf(z,2)) then return z
else errach list('spur0,n,i,v1,v2,v3)>>
end;
symbolic procedure spurdim(u,i,l,v,n);
begin scalar w,x,y,z,z1; integer m;
a: if null u
then return if null v then n
else if flagp(l,'nospur) then multfs(n,mkgamf(v,l))
else multfs(n,sprgen v)
else if not(car u memq cdr u)
then <<v := car u . v; u := cdr u; go to a>>;
x := car u;
y := cdr u;
w := y;
m := 1;
b: if x memq i then go to d
else if not(x eq car w) then go to c
else if null(w := mkdot(x,x)) then return z;
if x memq i then w := ndims!*;
return addfs(mkf(w,spurdim(delete(x,y),i,l,v,n)),z);
c: z1 := mkdot(x,car w);
if car w memq i
then z := addfs(spurdim(subst(x,car w,remove(y,m)),
i,l,v,2*n),z)
else if z1
then z := addfs(mkf(z1,spurdim(remove(y,m),i,l,v,2*n)),z);
w := cdr w;
n := -n;
m := m+1;
go to b;
d: while not(x eq car w) do
<<z:= addfs(spurdim(subst(car w,x,remove(y,m)),i,l,v,2*n),z);
w := cdr w;
n := -n;
m := m+1>>;
return addfs(mkf(ndims!*,spurdim(delete(x,y),i,l,v,n)),z)
end;
symbolic procedure appn(u,n);
if n=1 then u else append(u,appn(u,n-1));
symbolic procedure other(u,v);
if u eq car v then cdr v else car v;
symbolic procedure kahane(u,i,l);
% The Kahane algorithm for Dirac matrix string reduction.
% Ref: Kahane, J., Journ. Math. Phys. 9 (1968) 1732-1738.
begin scalar p,r,v,w,x,y,z; integer k,m;
k := 0;
% mark:
if eqcar(u,'a) then go to a1;
a: p := not p; % Vector parity.
if null u then go to d else if car u member i then go to c;
a1: w := aconc!*(w,car u);
b: u := cdr u;
go to a;
c: y := car u . p;
z := (x . (y . w)) . z;
x := y;
w := nil;
k := k+1;
go to b;
d: z := (nil . (x . w)) . z;
% Beware ... end of string has opposite convention.
% pass2:
m := 1;
l1: if null z then go to l9;
u := caar z;
x := cadar z;
w := cddar z;
z := cdr z;
m := m+1;
if null u then go to l2
else if (car u eq car x) and exc(x,cdr u) then go to l7;
w := reverse w;
r := t;
l2: p := not exc(x,r);
x := car x;
y := nil;
l3: if null z
then rerror(hephys,11,"Unmatched index" .
if y then if not atom cadar y then cadar y
else if not atom caar y then caar y
else nil
else nil)
else if (x eq car (i := cadar z)) and not exc(i,p)
then go to l5
else if (x eq car (i := caar z)) and exc(i,p) then go to l4;
y := car z . y;
z := cdr z;
go to l3;
l4: x := cadar z;
w := appr(cddar z,w);
r := t;
go to l6;
l5: x := caar z;
w := append(cddar z,w);
r := nil;
l6: z := appr(y,cdr z);
if null x then go to l8
else if not eqcar(u,car x) then go to l2;
l7: if w and cdr u then w := aconc!*(cdr w,car w);
v := multfs(brace(w,l,nil),v); % v := ('brace . l . w) . v;
go to l1;
l8: v := mkg(w,l); % v := list('g . l . w);
z := reverse z;
k := k/2;
go to l1;
l9: u := 2**k;
if not evenp(k-m) then u := - u;
return multd!*(u,v) % return 'times . u . v;
end;
symbolic procedure appr(u,v);
if null u then v else appr(cdr u,car u . v);
symbolic procedure exc(u,v);
if null cdr u then v else not v;
symbolic procedure brace(u,l,i);
if null u then 2
else if xnp(i,u) or flagp(l,'nospur)
then addf(mkg1(u,l),mkg1(reverse u,l))
else if car u eq 'a
then if hevenp u then addfs(mkg(u,l),
negf mkg('a . reverse cdr u,l))
else mkf(mka l,spr2(cdr u,l,2,nil))
else if hevenp u then spr2(u,l,2,nil)
else spr1(u,l,2,nil);
symbolic procedure spr1(u,l,n,b);
if null u then nil
else if null cdr u then multd!*(n,mkg1(u,l))
else begin scalar m,x,z;
x := u;
m := 1;
a: if null x then return z;
z:= addfs(mkf(mkg1(list car x,l),
if null b then spurr(remove(u,m),l,nil,n)
else spr1(remove(u,m),l,n,nil)),
z);
x := cdr x;
n := - n;
m := m+1;
go to a
end;
symbolic procedure spr2(u,l,n,b);
if null cddr u and null b then multd!*(n,mkdot(car u,cadr u))
else (lambda x; if b then addfs(spr1(u,l,n,b),x) else x)
addfs(spurr(u,l,nil,n),
mkf(mka l,spurr(append(u,list 'a),l,nil,n)));
symbolic procedure hevenp u;
null u or not hevenp cdr u;
symbolic procedure bassoc(u,v);
if null v then nil
else if u eq caar v or u eq cdar v then car v
else bassoc(u,cdr v);
symbolic procedure memlis(u,v);
if null v then nil
else if u member car v then car v
else memlis(u,cdr v);
symbolic procedure spurr(u,l,v,n);
begin scalar w,x,y,z,z1; integer m;
a: if null u then go to b
else if car u member cdr u then go to g;
v := car u . v;
u := cdr u;
go to a;
b: return if null v then n
else if flagp(l,'nospur) then multd!*(n,mkgamf(v,l))
else multd!*(n,sprgen v);
g: x := car u;
y := cdr u;
w := y;
m := 1;
h: if not(x eq car w) then go to h1
else if null(w:= mkdot(x,x)) then return z
else return addfs(mkf(w,spurr(delete(x,y),l,v,n)),z);
h1: z1 := mkdot(x,car w);
if z1 then z:= addfs(mkf(z1,spurr(remove(y,m),l,v,2*n)),z);
w := cdr w;
n := - n;
m := m+1;
go to h
end;
symbolic procedure sprgen v;
begin scalar x,y,z;
if not (car v eq 'a) then return sprgen1(v,t)
else if null (x := comb(v := cdr v,4)) then return nil
else if null cdr x then go to e;
c: if null x then return multpf('i to 1,z);
y := mkepsf car x;
if asign(car x,v,1)=-1 then y := negf y;
z := addf(multf(y,sprgen1(setdiff(v,car x),t)),z);
d: x := cdr x;
go to c;
e: z := mkepsf car x;
go to d
end;
symbolic procedure asign(u,v,n);
if null u then n else asign(cdr u,v,asign1(car u,v,-1)*n);
symbolic procedure asign1(u,v,n);
if u eq car v then n else asign1(u,cdr v,-n);
symbolic procedure sprgen1(u,b);
if null u then nil
else if null cddr u then (lambda x; if b then x else negf x)
mkdot(car u,cadr u)
else begin scalar w,x,y,z;
x := car u;
u := cdr u;
y := u;
a: if null u then return z
else if null(w:= mkdot(x,car u)) then go to c;
z := addf(multf(w,sprgen1(delete(car u,y),b)),z);
c: b := not b;
u := cdr u;
go to a
end;
% ****************** FUNCTIONS FOR EPSILON ALGEBRA ******************
put('eps,'simpfn,'simpeps);
symbolic procedure mkepsf u;
(lambda x; (lambda y; if null car x then negf y else y) mksf cdr x)
mkepsk u;
symbolic procedure esum(u,i,v,w,x);
begin scalar y,z,z1;
z := car u;
u := cdr u;
if cdr z neq 1
then u := multf(exptf(mkepsf cdar z,cdr z-1),u);
z := cdar z;
a: if repeats z then return nil;
b: if null z then return isimp1(u,i,v,reverse y . w,x)
else if car z member i
then <<if z1 := bassoc(car z,v)
then <<v := delete(z1,v);
i := delete(car z,i);
z := append(reverse y,other(car z,z1) . cdr z);
y := nil;
go to a>>
else if z1 := memlis(car z,w)
then <<z := append(reverse y,z);
y := intersection(i,intersection(z,z1));
return isimp1(multfs(emult1(z1,z,y),u),
setdiff(i,y),
v,
delete(z1,w),
x)>>>>;
y := car z . y;
z := cdr z;
go to b
end;
symbolic procedure emult u;
if null cdr u then mkepsf car u
else if null cddr u then emult1(car u,cadr u,nil)
else multfs(emult1(car u,cadr u,nil),emult cddr u);
symbolic procedure emult1(u,v,i);
(lambda (x,y);
(lambda (m,n);
if m=4 then 24*n
else if m=3 then multd(6*n,mkdot(car x,car y))
else multd!*(n*(if m = 0 then 1 else m),
car detq maplist(x,
function (lambda k;
maplist(y,
function (lambda j;
mkdot(car k,car j) . 1))))))
(length i,
(lambda j; nb if permp(u,append(i,x)) then not j else j)
permp(v,append(i,y))))
(setdiff(u,i),setdiff(v,i));
endmodule;
end;