* Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaL, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaR, tensor)
//////////
// default ctor, dtor, copy ctor, assignment operator and helpers
DEFAULT_CTORS(diracone)
DEFAULT_CTORS(diracgamma)
DEFAULT_CTORS(diracgamma5)
+DEFAULT_CTORS(diracgammaL)
+DEFAULT_CTORS(diracgammaR)
//////////
// other constructors
DEFAULT_ARCHIVING(diracone)
DEFAULT_ARCHIVING(diracgamma)
DEFAULT_ARCHIVING(diracgamma5)
+DEFAULT_ARCHIVING(diracgammaL)
+DEFAULT_ARCHIVING(diracgammaR)
//////////
// functions overriding virtual functions from base classes
void clifford::print(const print_context & c, unsigned level) const
{
- if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
+ if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgammaL>(seq[0]) &&
+ !is_a<diracgammaR>(seq[0]) && !is_a<diracgamma>(seq[0]) &&
+ !is_a<diracone>(seq[0])) {
// dirac_slash() object is printed differently
if (is_a<print_tree>(c))
DEFAULT_COMPARE(diracone)
DEFAULT_COMPARE(diracgamma)
DEFAULT_COMPARE(diracgamma5)
+DEFAULT_COMPARE(diracgammaL)
+DEFAULT_COMPARE(diracgammaR)
DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
+DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
+DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
static void base_and_index(const ex & c, ex & b, ex & i)
if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
i = c.op(1);
b = _ex1;
- } else if (is_a<diracgamma5>(c.op(0))) { // gamma5
+ } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
i = _ex0;
b = _ex1;
} else { // slash object, generate new dummy index
GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
unsigned char rl = ex_to<clifford>(*self).get_representation_label();
+ ex dim = ex_to<idx>(self->op(1)).get_dim();
+ if (other->nops() > 1)
+ dim = minimal_dim(dim, ex_to<idx>(self->op(1)).get_dim());
+
if (is_a<clifford>(*other)) {
// Contraction only makes sense if the represenation labels are equal
if (ex_to<clifford>(*other).get_representation_label() != rl)
return false;
- ex dim = ex_to<idx>(self->op(1)).get_dim();
-
// gamma~mu gamma.mu = dim ONE
if (other - self == 1) {
*self = dim;
*other = _ex1;
return true;
}
+
+ } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
+
+ // x.mu gamma~mu -> x-slash
+ *self = dirac_slash(other->op(0), dim, rl);
+ *other = _ex1;
+ return true;
}
return false;
}
/** Perform automatic simplification on noncommutative product of clifford
- * objects. This removes superfluous ONEs, permutes gamma5's to the front
+ * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
* and removes squares of gamma objects. */
ex clifford::simplify_ncmul(const exvector & v) const
{
bool something_changed = false;
int sign = 1;
- // Anticommute gamma5's to the front
+ // Anticommute gamma5/L/R's to the front
if (s.size() >= 2) {
exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
while (true) {
exvector::iterator it = next_to_last;
while (true) {
exvector::iterator it2 = it + 1;
- if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !is_a<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
- it->swap(*it2);
- sign = -sign;
- something_changed = true;
+ if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
+ ex e1 = it->op(0), e2 = it2->op(0);
+
+ if (is_a<diracgamma5>(e2)) {
+
+ if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
+
+ // gammaL/R gamma5 -> gamma5 gammaL/R
+ it->swap(*it2);
+ something_changed = true;
+
+ } else if (!is_a<diracgamma5>(e1)) {
+
+ // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
+ // x gamma5 -> -gamma5 x
+ it->swap(*it2);
+ sign = -sign;
+ something_changed = true;
+ }
+
+ } else if (is_a<diracgammaL>(e2)) {
+
+ if (is_a<diracgammaR>(e1)) {
+
+ // gammaR gammaL -> 0
+ return _ex0;
+
+ } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
+
+ // gammaL gammaL -> gammaL gammaL (do nothing)
+ // gamma5 gammaL -> gamma5 gammaL (do nothing)
+ // x gammaL -> gammaR x
+ it->swap(*it2);
+ *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
+ something_changed = true;
+ }
+
+ } else if (is_a<diracgammaR>(e2)) {
+
+ if (is_a<diracgammaL>(e1)) {
+
+ // gammaL gammaR -> 0
+ return _ex0;
+
+ } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
+
+ // gammaR gammaR -> gammaR gammaR (do nothing)
+ // gamma5 gammaR -> gamma5 gammaR (do nothing)
+ // x gammaR -> gammaL x
+ it->swap(*it2);
+ *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
+ something_changed = true;
+ }
+ }
}
if (it == first)
break;
ex & b = it[1];
if (!is_a<clifford>(a) || !is_a<clifford>(b))
continue;
- bool a_is_diracgamma = is_a<diracgamma>(a.op(0));
- bool b_is_diracgamma = is_a<diracgamma>(b.op(0));
+
+ const ex & ag = a.op(0);
+ const ex & bg = b.op(0);
+ bool a_is_diracgamma = is_a<diracgamma>(ag);
+ bool b_is_diracgamma = is_a<diracgamma>(bg);
+
if (a_is_diracgamma && b_is_diracgamma) {
+
const ex & ia = a.op(1);
const ex & ib = b.op(1);
if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
b = dirac_ONE(representation_label);
something_changed = true;
}
- } else if (is_a<diracgamma5>(a.op(0)) && is_a<diracgamma5>(b.op(0))) {
+
+ } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
+
// Remove squares of gamma5
a = dirac_ONE(representation_label);
b = dirac_ONE(representation_label);
something_changed = true;
- } else if (!a_is_diracgamma && !b_is_diracgamma) {
- const ex & ba = a.op(0);
- const ex & bb = b.op(0);
- if (ba.is_equal(bb)) { // a\ a\ -> a^2
- varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
- a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
- b = dirac_ONE(representation_label);
- something_changed = true;
- }
+
+ } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
+ || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
+
+ // Remove squares of gammaL/R
+ b = dirac_ONE(representation_label);
+ something_changed = true;
+
+ } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
+
+ // gammaL and gammaR are orthogonal
+ return _ex0;
+
+ } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
+
+ // gamma5 gammaL -> -gammaL
+ a = dirac_ONE(representation_label);
+ sign = -sign;
+ something_changed = true;
+
+ } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
+
+ // gamma5 gammaR -> gammaR
+ a = dirac_ONE(representation_label);
+ something_changed = true;
+
+ } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
+
+ // a\ a\ -> a^2
+ varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
+ a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
+ b = dirac_ONE(representation_label);
+ something_changed = true;
}
}
}
return clifford(diracgamma5(), rl);
}
+ex dirac_gammaL(unsigned char rl)
+{
+ return clifford(diracgammaL(), rl);
+}
+
+ex dirac_gammaR(unsigned char rl)
+{
+ return clifford(diracgammaR(), rl);
+}
+
ex dirac_gamma6(unsigned char rl)
{
return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
{
if (is_a<clifford>(e)) {
- if (ex_to<clifford>(e).get_representation_label() == rl
- && is_a<diracone>(e.op(0)))
+ if (!ex_to<clifford>(e).get_representation_label() == rl)
+ return _ex0;
+ const ex & g = e.op(0);
+ if (is_a<diracone>(g))
return trONE;
+ else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
+ return trONE/2;
else
return _ex0;
if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
return _ex0;
- // Expand product, if necessary
- ex e_expanded = e.expand();
+ // Substitute gammaL/R and expand product, if necessary
+ ex e_expanded = e.subs(lst(
+ dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
+ dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
+ )).expand();
if (!is_a<ncmul>(e_expanded))
return dirac_trace(e_expanded, rl, trONE);
return _ex0;
// Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
+ // (the epsilon is always 4-dimensional)
if (num == 5) {
ex b1, i1, b2, i2, b3, i3, b4, i4;
base_and_index(e.op(1), b1, i1);
base_and_index(e.op(2), b2, i2);
base_and_index(e.op(3), b3, i3);
base_and_index(e.op(4), b4, i4);
- return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed();
+ return trONE * I * (lorentz_eps(ex_to<idx>(i1).replace_dim(_ex4), ex_to<idx>(i2).replace_dim(_ex4), ex_to<idx>(i3).replace_dim(_ex4), ex_to<idx>(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed();
}
// Tr gamma5 S_2k =
// I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
+ // (the epsilon is always 4-dimensional)
exvector ix(num-1), bv(num-1);
for (unsigned i=1; i<num; i++)
base_and_index(e.op(i), bv[i-1], ix[i-1]);
v.push_back(ix[n]);
}
int sign = permutation_sign(iv, iv + num);
- result += sign * eps0123(idx1, idx2, idx3, idx4)
+ result += sign * lorentz_eps(ex_to<idx>(idx1).replace_dim(_ex4), ex_to<idx>(idx2).replace_dim(_ex4), ex_to<idx>(idx3).replace_dim(_ex4), ex_to<idx>(idx4).replace_dim(_ex4))
* trace_string(v.begin(), num - 4);
}
}
// Stupid recursive bubble sort because we only want to swap adjacent gammas
exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
- if (is_a<diracgamma5>(it->op(0)))
+ if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
++it;
while (it != next_to_last) {
if (it[0].compare(it[1]) > 0) {
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