From: Christian Bauer <Christian.Bauer@uni-mainz.de>
Date: Wed, 25 Apr 2001 20:41:13 +0000 (+0000)
Subject: dirac_trace() can handle arbitrary gamma strings without gamma5. The only
X-Git-Tag: release_0-8-3~33
X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=commitdiff_plain;h=53604c3e255c392fb31af8bcecf57f2918495d28;ds=sidebyside

dirac_trace() can handle arbitrary gamma strings without gamma5. The only
remaining unimplemented case is now Tr[gamma5 S_2n]
---

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index eb1629d4..d346c44a 100644
--- a/ginac/clifford.cpp
+++ b/ginac/clifford.cpp
@@ -363,6 +363,28 @@ ex dirac_trace(const ex & e, unsigned char rl = 0)
 			// Tr gamma_mu gamma_nu = 4 g_mu_nu
 			if (num == 2)
 				return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
+
+			// Traces of 4 or more gammas are computed recursively:
+			// Tr gamma_mu1 gamma_mu2 ... gamma_mun =
+			//   + eta_mu1_mu2 * Tr gamma_mu3 ... gamma_mun
+			//   - eta_mu1_mu3 * Tr gamma_mu2 gamma_mu4 ... gamma_mun
+			//   + eta_mu1_mu4 * Tr gamma_mu3 gamma_mu3 gamma_mu5 ... gamma_mun
+			//   - ...
+			//   + eta_mu1_mun * Tr gamma_mu2 ... gamma_mu(n-1)
+			exvector v(num - 2);
+			int sign = 1;
+			const ex &ix1 = e.op(0).op(1);
+			ex result;
+			for (int i=1; i<num; i++) {
+				for (int n=1, j=0; n<num; n++) {
+					if (n == i)
+						continue;
+					v[j++] = e.op(n);
+				}
+				result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
+				sign = -sign;
+			}
+			return result;
 		}
 
 		throw (std::logic_error("dirac_trace: don't know how to compute trace"));
diff --git a/ginac/clifford.h b/ginac/clifford.h
index 4d771530..f1783396 100644
--- a/ginac/clifford.h
+++ b/ginac/clifford.h
@@ -126,7 +126,10 @@ ex dirac_gamma(const ex & mu, unsigned char rl = 0);
 ex dirac_gamma5(unsigned char rl = 0);
 
 /** Calculate the trace of an expression containing gamma objects with
- *  a specified representation label.
+ *  a specified representation label. The computed trace is a linear
+ *  functional that is equal to the usual trace only in D = 4 dimensions.
+ *  In particular, the functional is non-cyclic in D != 4 dimensions when
+ *  gamma5 is involved.
  *
  *  @param rl Representation label */
 ex dirac_trace(const ex & e, unsigned char rl = 0);