def test_gamma_matrix_class():
i, j, k = tensor_indices('i,j,k', G.Lorentz)
# define another type of TensorHead to see if exprs are correctly handled:
A = tensorhead('A', [G.Lorentz], [[1]])
t = A(k) * G(i) * G(-i)
ts = simplify(t)
assert ts == 4 * A(k)
t = G(i) * A(k) * G(j)
ts = simplify(t)
assert ts == A(k) * G(i) * G(j)
execute_gamma_simplify_tests_for_function(simplify, D=4)
def test_simple_trace_cases_symbolic_dim():
from sympy import symbols
D = symbols('D')
G = GammaMatrixHead(dim=D)
m0, m1, m2, m3 = tensor_indices('m0:4', G.LorentzIndex)
g = G.LorentzIndex.metric
t = G(m0) * G(m1)
t1 = G._trace_single_line(t)
assert _is_tensor_eq(t1, 4 * G.LorentzIndex.metric(m0, m1))
t = G(m0) * G(m1) * G(m2) * G(m3)
t1 = G._trace_single_line(t)
t2 = -4 * g(m0, m2) * g(m1, m3) + 4 * g(m0, m1) * g(m2, m3) + 4 * g(
m0, m3) * g(m1, m2)
assert _is_tensor_eq(t1, t2)
def test_gamma_matrix_class():
i, j, k = tensor_indices('i,j,k', G.LorentzIndex)
# define another type of TensorHead to see if exprs are correctly handled:
A = tensorhead('A', [G.LorentzIndex], [[1]])
t = A(k) * G(i) * G(-i)
ts = simplify(t)
assert _is_tensor_eq(
ts, 4 * A(k) * DiracSpinorIndex.delta(DiracSpinorIndex.auto_left,
-DiracSpinorIndex.auto_right))
t = G(i) * A(k) * G(j)
ts = simplify(t)
assert _is_tensor_eq(ts, A(k) * G(i) * G(j))
execute_gamma_simplify_tests_for_function(simplify, D=4)
def test_simple_trace_cases_symbolic_dim():
from sympy import symbols
D = symbols('D')
G = GammaMatrixHead(dim=D)
m0, m1, m2, m3 = tensor_indices('m0:4', G.Lorentz)
g = G.Lorentz.metric
t = G(m0) * G(m1)
t1 = G.trace_tens(t)
assert t1 == 4 * G.Lorentz.metric(m0, m1)
t = G(m0) * G(m1) * G(m2) * G(m3)
t1 = G.trace_tens(t)
t2 = -4 * g(m0, m2) * g(m1, m3) + 4 * g(m0, m1) * g(m2, m3) + 4 * g(
m0, m3) * g(m1, m2)
assert t1 == t2
def test_gamma_matrix_class():
i, j, k = tensor_indices("i,j,k", LorentzIndex)
# define another type of TensorHead to see if exprs are correctly handled:
A = TensorHead("A", [LorentzIndex])
t = A(k) * G(i) * G(-i)
ts = simplify_gamma_expression(t)
assert _is_tensor_eq(
ts, Matrix([[4, 0, 0, 0], [0, 4, 0, 0], [0, 0, 4, 0], [0, 0, 0, 4]]) * A(k)
)
t = G(i) * A(k) * G(j)
ts = simplify_gamma_expression(t)
assert _is_tensor_eq(ts, A(k) * G(i) * G(j))
execute_gamma_simplify_tests_for_function(simplify_gamma_expression, D=4)
def test_simplify_lines():
i0, i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, i11, i12 = tensor_indices(
'i0:13', G.LorentzIndex)
s0,s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14,s15,s16 = \
tensor_indices('s0:17', DiracSpinorIndex)
g = G.LorentzIndex.metric
Sdelta = DiracSpinorIndex.delta
t = G(i1, s1, -s2) * G(i2, s2, -s1) * G(i3, s4, -s5) * G(i4, s5, -s6) * G(
i5, s7, -s8)
r = G.simplify_lines(t)
assert r.equals(4 * G(i5, s7, -s8) * G(i3, s4, -s0) * G(i4, s0, -s6) *
g(i1, i2))
t = G(i1, s1, -s2) * G(i2, s2, -s1) * G(i3, s4, -s5) * G(-i3, s5, -s6) * G(
i5, s7, -s8)
r = G.simplify_lines(t)
assert r.equals(16 * G(i5, s7, -s8) * Sdelta(s4, -s6) * g(i1, i2))
t = G(i1, s1, -s2) * G(i2, s2, -s1) * G(i3, s4, -s5) * G(i4, s5, -s6) * G(
i5, s7, -s8)
r = G.simplify_lines(t)
assert r.equals(4 * G(i5, s7, -s8) * G(i3, s4, s0) * G(i4, -s0, -s6) *
g(i1, i2))
t = G(i5, s7, -s8) * G(i6, s9, -s10) * G(i1, s1, -s2) * G(i3, s4, -s5) * G(
i2, s2, -s1) * G(i4, s5, -s6) * G(-i6, s10, -s9)
r = G.simplify_lines(t)
assert r.equals(64 * G(i5, s7, -s8) * G(i3, s4, s0) * G(i4, -s0, -s6) *
g(i1, i2))
t = G(i5,s7,-s8)*G(i6,s9,-s10)*G(i1,s1,-s2)*G(i7,s12,-s11)*G(i3,s4,-s5)*\
G(i2,s2,-s1)*G(i4,s5,-s6)*G(-i6,s10,-s9)*G(-i7,s11,-s13)
r = G.simplify_lines(t)
assert r.equals(256*G(i5, s7, -s8)*G(i3, s4, s0)*G(i4, -s0, -s6)*\
g(i1, i2)*Sdelta(s12,-s13))
def test_get_lines():
i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13 = \
tensor_indices('i0:14', G.LorentzIndex)
s0,s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14,s15,s16 = \
tensor_indices('s0:17', DiracSpinorIndex)
t = G(i1, s1, -s2) * G(i2, s3, -s4) * G(i4, s2, -s6) * G(i3, s4, -s3)
r = get_lines(t, DiracSpinorIndex)
assert r == ([[0, 2]], [[1, 3]], [])
t = G(i1,s1,-s2)*G(i2,s2,-s3)*G(i3,s3,-s4)*G(i4,s4,-s5)*\
G(i5,s6,-s7)*G(i6,s7,-s8)*G(i7,s8,-s9)*G(i8,s9,-s6)
r = get_lines(t, DiracSpinorIndex)
assert r == ([[0, 1, 2, 3]], [[4, 5, 6, 7]], [])
t = G(i1,s1,-s2)*G(i0,s0,-s10)*G(i2,s2,-s3)*G(i3,s3,-s4)*\
G(i4,s4,-s5)*G(i5,s6,-s7)*G(i6,s7,-s8)*G(i7,s8,-s9)*\
G(i8,s9,-s6)*G(i9,s10,-s0)
r = get_lines(t, DiracSpinorIndex)
assert r == ([[0, 2, 3, 4]], [[5, 6, 7, 8], [1, 9]], [])
t = G(i1,s1,-s2)*G(i11,s12,-s13)*G(i0,s0,-s10)*G(i2,s2,-s3)*G(i3,s3,-s4)*\
G(i4,s4,-s5)*G(i5,s6,-s7)*G(i10,s11,-s12)*G(i6,s7,-s8)*G(i7,s8,-s9)*\
G(i8,s9,-s6)*G(i9,s10,-s0)
r = get_lines(t, DiracSpinorIndex)
assert r == ([[7, 1], [0, 3, 4, 5]], [[6, 8, 9, 10], [2, 11]], [])
t = G(i4,s4,-s5)*G(i5,s6,-s7)*G(i10,s11,-s12)*G(i6,s7,-s8)*G(i7,s8,-s9)*\
G(i8,s9,-s6)*G(i9,s10,-s0)*\
G(i1,s1,-s2)*G(i11,s12,-s13)*G(i0,s0,-s10)*G(i2,s2,-s3)*G(i3,s3,-s4)
r = get_lines(t, DiracSpinorIndex)
assert r == ([[7, 10, 11, 0], [2, 8]], [[1, 3, 4, 5], [6, 9]], [])
t = G(i8,s9,-s6)*G(i9,s10,-s0)*G(i4,s4,-s5)*G(i13,s14,-s15)*\
G(i10,s11,-s12)*G(i1,s1,-s2)*G(i11,s12,-s13)*\
G(i0,s0,-s10)*G(i6,s7,-s8)*G(i7,s8,-s9)*\
G(i2,s2,-s3)*G(i12,s13,-s14)*G(i3,s3,-s4)*G(i5,s6,-s7)
r = get_lines(t, DiracSpinorIndex)
assert r == ([[5, 10, 12, 2], [4, 6, 11, 3]], [[1, 7], [0, 13, 8, 9]], [])
def g(xx, yy):
return (G(xx) * G(yy) + G(yy) * G(xx)) / 2
def test_gamma_matrix_trace():
gamma_trace = G.gamma_trace
g = G.LorentzIndex.metric
m0, m1, m2, m3, m4, m5, m6 = tensor_indices('m0:7', G.LorentzIndex)
n0, n1, n2, n3, n4, n5 = tensor_indices('n0:6', G.LorentzIndex)
# working in D=4 dimensions
D = 4
# traces of odd number of gamma matrices are zero:
t = G(m0)
t1 = gamma_trace(t)
assert t1.equals(0)
t = G(m0) * G(m1) * G(m2)
t1 = gamma_trace(t)
assert t1.equals(0)
t = G(m0) * G(m1) * G(-m0)
t1 = gamma_trace(t)
assert t1.equals(0)
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4)
t1 = gamma_trace(t)
assert t1.equals(0)
# traces without internal contractions:
t = G(m0) * G(m1)
t1 = gamma_trace(t)
assert _is_tensor_eq(t1, 4 * g(m0, m1))
t = G(m0) * G(m1) * G(m2) * G(m3)
t1 = gamma_trace(t)
t2 = -4 * g(m0, m2) * g(m1, m3) + 4 * g(m0, m1) * g(m2, m3) + 4 * g(
m0, m3) * g(m1, m2)
st2 = str(t2)
assert _is_tensor_eq(t1, t2)
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(m5)
t1 = gamma_trace(t)
t2 = t1 * g(-m0, -m5)
t2 = t2.contract_metric(g)
assert _is_tensor_eq(t2, D * gamma_trace(G(m1) * G(m2) * G(m3) * G(m4)))
# traces of expressions with internal contractions:
t = G(m0) * G(-m0)
t1 = gamma_trace(t)
assert t1.equals(4 * D)
t = G(m0) * G(m1) * G(-m0) * G(-m1)
t1 = gamma_trace(t)
assert t1.equals(8 * D - 4 * D**2)
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(-m0)
t1 = gamma_trace(t)
t2 = (-4*D)*g(m1, m3)*g(m2, m4) + (4*D)*g(m1, m2)*g(m3, m4) + \
(4*D)*g(m1, m4)*g(m2, m3)
assert t1.equals(t2)
t = G(-m5) * G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(-m0) * G(m5)
t1 = gamma_trace(t)
t2 = (32*D + 4*(-D + 4)**2 - 64)*(g(m1, m2)*g(m3, m4) - \
g(m1, m3)*g(m2, m4) + g(m1, m4)*g(m2, m3))
assert t1.equals(t2)
t = G(m0) * G(m1) * G(-m0) * G(m3)
t1 = gamma_trace(t)
assert t1.equals((-4 * D + 8) * g(m1, m3))
# p, q = S1('p,q')
# ps = p(m0)*G(-m0)
# qs = q(m0)*G(-m0)
# t = ps*qs*ps*qs
# t1 = gamma_trace(t)
# assert t1 == 8*p(m0)*q(-m0)*p(m1)*q(-m1) - 4*p(m0)*p(-m0)*q(m1)*q(-m1)
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(m5) * G(-m0) * G(-m1) * G(
-m2) * G(-m3) * G(-m4) * G(-m5)
t1 = gamma_trace(t)
assert t1.equals(-4 * D**6 + 120 * D**5 - 1040 * D**4 + 3360 * D**3 -
4480 * D**2 + 2048 * D)
t = G(m0) * G(m1) * G(n1) * G(m2) * G(n2) * G(m3) * G(m4) * G(-n2) * G(
-n1) * G(-m0) * G(-m1) * G(-m2) * G(-m3) * G(-m4)
t1 = gamma_trace(t)
tresu = -7168 * D + 16768 * D**2 - 14400 * D**3 + 5920 * D**4 - 1232 * D**5 + 120 * D**6 - 4 * D**7
assert t1.equals(tresu)
# checked with Mathematica
# In[1]:= <<Tracer.m
# In[2]:= Spur[l];
# In[3]:= GammaTrace[l, {m0},{m1},{n1},{m2},{n2},{m3},{m4},{n3},{n4},{m0},{m1},{m2},{m3},{m4}]
t = G(m0) * G(m1) * G(n1) * G(m2) * G(n2) * G(m3) * G(m4) * G(n3) * G(
n4) * G(-m0) * G(-m1) * G(-m2) * G(-m3) * G(-m4)
t1 = gamma_trace(t)
# t1 = t1.expand_coeff()
c1 = -4 * D**5 + 120 * D**4 - 1200 * D**3 + 5280 * D**2 - 10560 * D + 7808
c2 = -4 * D**5 + 88 * D**4 - 560 * D**3 + 1440 * D**2 - 1600 * D + 640
assert _is_tensor_eq(t1, c1*g(n1, n4)*g(n2, n3) + c2*g(n1, n2)*g(n3, n4) + \
(-c1)*g(n1, n3)*g(n2, n4))
p, q = tensorhead('p,q', [G.LorentzIndex], [[1]])
ps = p(m0) * G(-m0)
qs = q(m0) * G(-m0)
p2 = p(m0) * p(-m0)
q2 = q(m0) * q(-m0)
pq = p(m0) * q(-m0)
t = ps * qs * ps * qs
r = gamma_trace(t)
assert _is_tensor_eq(r, 8 * pq * pq - 4 * p2 * q2)
t = ps * qs * ps * qs * ps * qs
r = gamma_trace(t)
assert r.equals(-12 * p2 * pq * q2 + 16 * pq * pq * pq)
t = ps * qs * ps * qs * ps * qs * ps * qs
r = gamma_trace(t)
assert r.equals(-32 * pq * pq * p2 * q2 + 32 * pq * pq * pq * pq +
4 * p2 * p2 * q2 * q2)
t = 4 * p(m1) * p(m0) * p(-m0) * q(-m1) * q(m2) * q(-m2)
assert _is_tensor_eq(gamma_trace(t), t)
t = ps * ps * ps * ps * ps * ps * ps * ps
r = gamma_trace(t)
assert r.equals(4 * p2 * p2 * p2 * p2)
def test_kahane_simplify1():
i0, i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, i11, i12, i13, i14, i15 = tensor_indices(
'i0:16', G.LorentzIndex)
mu, nu, rho, sigma = tensor_indices("mu, nu, rho, sigma", G.LorentzIndex)
KD = DiracSpinorIndex.delta
sl = DiracSpinorIndex.auto_left
sr = DiracSpinorIndex.auto_right
D = 4
s0,s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14,s15,s16 = \
tensor_indices('s0:17', DiracSpinorIndex)
t = DiracSpinorIndex.delta(s0, s1)
t = G(i0) * G(i1)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(t)
t = G(i0) * G(i1) * G(-i0)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(-2 * G(i1))
t = G(i0, s0, -s1) * G(i1, s1, -s2) * G(-i0, s2, -s3)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(-2 * G(i1, s0, -s3))
t = G(i0, s0, -s1) * G(i1, s1, -s2)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(t)
t = G(i0, s0, -s1) * G(i1, s1, -s0)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(t)
t = G(i0) * G(-i0)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(4 * KD(sl, -sr))
t = G(i0, s0, -s1) * G(-i0, s1, -s2)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(4 * KD(s0, -s2))
t = G(i0, s0, -s1) * G(-i0, s1, -s0)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(16)
t = G(i0) * G(i1) * G(-i0)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(-2 * G(i1))
t = G(i0) * G(i1) * G(-i0) * G(-i1)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals((2 * D - D**2) * KD(sl, -sr))
t = G(i0, s0, -s1) * G(i1, s1, -s2) * G(-i0, s2, -s3) * G(-i1, s3, -s0)
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(4 * (2 * D - D**2))
t = G(i0, s0, -s1) * G(-i0, s2, -s3) * G(i1, s1, -s2) * G(-i1, s3, -s0)
raises(ValueError, lambda: G._kahane_simplify(t.coeff, t._tids))
t = (G(mu) * G(nu) * G(-nu) * G(-mu))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(D**2 * KD(sl, -sr))
t = (G(mu, s0, -s1) * G(nu, s1, -s2) * G(-nu, s2, -s3) * G(-mu, s3, -s4))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(D**2 * KD(s0, -s4))
t = (G(mu, s0, -s1) * G(nu, s1, -s2) * G(-nu, s2, -s3) * G(-mu, s3, -s0))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(4 * D**2)
t = (G(mu) * G(nu) * G(-rho) * G(-nu) * G(-mu) * G(rho))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals((4 * D - 4 * D**2 + D**3) * KD(sl, -sr))
t = (G(-mu) * G(-nu) * G(-rho) * G(-sigma) * G(nu) * G(mu) * G(sigma) *
G(rho))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals((-16 * D + 24 * D**2 - 8 * D**3 + D**4) * KD(sl, -sr))
t = (G(-mu) * G(nu) * G(-rho) * G(sigma) * G(rho) * G(-nu) * G(mu) *
G(-sigma))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals((8 * D - 12 * D**2 + 6 * D**3 - D**4) * KD(sl, -sr))
# Expressions with free indices:
t = (G(mu) * G(nu) * G(rho) * G(sigma) * G(-mu))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(-2 * G(sigma) * G(rho) * G(nu))
t = (G(mu, s0, -s1) * G(nu, s1, -s2) * G(rho, s2, -s3) *
G(sigma, s3, -s4) * G(-mu, s4, -s5))
r = G._kahane_simplify(t.coeff, t._tids)
assert r.equals(-2 * G(sigma, s0, -s1) * G(rho, s1, -s2) * G(nu, s2, -s5))
def execute_gamma_simplify_tests_for_function(tfunc, D):
"""
Perform tests to check if sfunc is able to simplify gamma matrix expressions.
Parameters
==========
`sfunc` a function to simplify a `TIDS`, shall return the simplified `TIDS`.
`D` the number of dimension (in most cases `D=4`).
"""
mu, nu, rho, sigma = tensor_indices("mu, nu, rho, sigma", G.LorentzIndex)
a1, a2, a3, a4, a5, a6 = tensor_indices("a1:7", G.LorentzIndex)
mu11, mu12, mu21, mu31, mu32, mu41, mu51, mu52 = tensor_indices(
"mu11, mu12, mu21, mu31, mu32, mu41, mu51, mu52", G.LorentzIndex)
mu61, mu71, mu72 = tensor_indices("mu61, mu71, mu72", G.LorentzIndex)
m0, m1, m2, m3, m4, m5, m6 = tensor_indices("m0:7", G.LorentzIndex)
def g(xx, yy):
return (G(xx) * G(yy) + G(yy) * G(xx)) / 2
# Some examples taken from Kahane's paper, 4 dim only:
if D == 4:
t = (G(a1) * G(mu11) * G(a2) * G(mu21) * G(-a1) * G(mu31) * G(-a2))
assert _is_tensor_eq(
tfunc(t),
-4 * G(mu11) * G(mu31) * G(mu21) - 4 * G(mu31) * G(mu11) * G(mu21))
t = (G(a1)*G(mu11)*G(mu12)*\
G(a2)*G(mu21)*\
G(a3)*G(mu31)*G(mu32)*\
G(a4)*G(mu41)*\
G(-a2)*G(mu51)*G(mu52)*\
G(-a1)*G(mu61)*\
G(-a3)*G(mu71)*G(mu72)*\
G(-a4))
assert _is_tensor_eq(tfunc(t), \
16*G(mu31)*G(mu32)*G(mu72)*G(mu71)*G(mu11)*G(mu52)*G(mu51)*G(mu12)*G(mu61)*G(mu21)*G(mu41) + 16*G(mu31)*G(mu32)*G(mu72)*G(mu71)*G(mu12)*G(mu51)*G(mu52)*G(mu11)*G(mu61)*G(mu21)*G(mu41) + 16*G(mu71)*G(mu72)*G(mu32)*G(mu31)*G(mu11)*G(mu52)*G(mu51)*G(mu12)*G(mu61)*G(mu21)*G(mu41) + 16*G(mu71)*G(mu72)*G(mu32)*G(mu31)*G(mu12)*G(mu51)*G(mu52)*G(mu11)*G(mu61)*G(mu21)*G(mu41))
# Fully G.Lorentz-contracted expressions, these return scalars:
def add_delta(ne):
return ne * DiracSpinorIndex.delta(DiracSpinorIndex.auto_left,
-DiracSpinorIndex.auto_right)
t = (G(mu) * G(-mu))
ts = add_delta(D)
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu) * G(nu) * G(-mu) * G(-nu))
ts = add_delta(2 * D - D**2) # -8
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu) * G(nu) * G(-nu) * G(-mu))
ts = add_delta(D**2) # 16
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu) * G(nu) * G(-rho) * G(-nu) * G(-mu) * G(rho))
ts = add_delta(4 * D - 4 * D**2 + D**3) # 16
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu) * G(nu) * G(rho) * G(-rho) * G(-nu) * G(-mu))
ts = add_delta(D**3) # 64
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1) * G(a2) * G(a3) * G(a4) * G(-a3) * G(-a1) * G(-a2) * G(-a4))
ts = add_delta(-8 * D + 16 * D**2 - 8 * D**3 + D**4) # -32
assert _is_tensor_eq(tfunc(t), ts)
t = (G(-mu) * G(-nu) * G(-rho) * G(-sigma) * G(nu) * G(mu) * G(sigma) *
G(rho))
ts = add_delta(-16 * D + 24 * D**2 - 8 * D**3 + D**4) # 64
assert _is_tensor_eq(tfunc(t), ts)
t = (G(-mu) * G(nu) * G(-rho) * G(sigma) * G(rho) * G(-nu) * G(mu) *
G(-sigma))
ts = add_delta(8 * D - 12 * D**2 + 6 * D**3 - D**4) # -32
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1) * G(a2) * G(a3) * G(a4) * G(a5) * G(-a3) * G(-a2) * G(-a1) *
G(-a5) * G(-a4))
ts = add_delta(64 * D - 112 * D**2 + 60 * D**3 - 12 * D**4 + D**5) # 256
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1) * G(a2) * G(a3) * G(a4) * G(a5) * G(-a3) * G(-a1) * G(-a2) *
G(-a4) * G(-a5))
ts = add_delta(64 * D - 120 * D**2 + 72 * D**3 - 16 * D**4 + D**5) # -128
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1) * G(a2) * G(a3) * G(a4) * G(a5) * G(a6) * G(-a3) * G(-a2) *
G(-a1) * G(-a6) * G(-a5) * G(-a4))
ts = add_delta(416 * D - 816 * D**2 + 528 * D**3 - 144 * D**4 + 18 * D**5 -
D**6) # -128
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1) * G(a2) * G(a3) * G(a4) * G(a5) * G(a6) * G(-a2) * G(-a3) *
G(-a1) * G(-a6) * G(-a4) * G(-a5))
ts = add_delta(416 * D - 848 * D**2 + 584 * D**3 - 172 * D**4 + 22 * D**5 -
D**6) # -128
assert _is_tensor_eq(tfunc(t), ts)
# Expressions with free indices:
t = (G(mu) * G(nu) * G(rho) * G(sigma) * G(-mu))
assert _is_tensor_eq(tfunc(t), (-2 * G(sigma) * G(rho) * G(nu) +
(4 - D) * G(nu) * G(rho) * G(sigma)))
t = (G(mu) * G(nu) * G(-mu))
assert _is_tensor_eq(tfunc(t), (2 - D) * G(nu))
t = (G(mu) * G(nu) * G(rho) * G(-mu))
assert _is_tensor_eq(
tfunc(t),
2 * G(nu) * G(rho) + 2 * G(rho) * G(nu) - (4 - D) * G(nu) * G(rho))
t = 2 * G(m2) * G(m0) * G(m1) * G(-m0) * G(-m1)
st = tfunc(t)
assert _is_tensor_eq(st, (D * (-2 * D + 4)) * G(m2))
t = G(m2) * G(m0) * G(m1) * G(-m0) * G(-m2)
st = tfunc(t)
assert _is_tensor_eq(st, ((-D + 2)**2) * G(m1))
t = G(m0) * G(m1) * G(m2) * G(m3) * G(-m1)
st = tfunc(t)
assert _is_tensor_eq(st, (D - 4) * G(m0) * G(m2) * G(m3) +
4 * G(m0) * g(m2, m3))
t = G(m0) * G(m1) * G(m2) * G(m3) * G(-m1) * G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, ((D - 4)**2) * G(m2) * G(m3) +
(8 * D - 16) * g(m2, m3))
t = G(m2) * G(m0) * G(m1) * G(-m2) * G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, ((-D + 2) * (D - 4) + 4) * G(m1))
t = G(m3) * G(m1) * G(m0) * G(m2) * G(-m3) * G(-m0) * G(-m2)
st = tfunc(t)
assert _is_tensor_eq(st, (-4 * D + (-D + 2)**2 * (D - 4) + 8) * G(m1))
t = 2 * G(m0) * G(m1) * G(m2) * G(m3) * G(-m0)
st = tfunc(t)
assert _is_tensor_eq(
st, ((-2 * D + 8) * G(m1) * G(m2) * G(m3) - 4 * G(m3) * G(m2) * G(m1)))
t = G(m5) * G(m0) * G(m1) * G(m4) * G(m2) * G(-m4) * G(m3) * G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, (((-D + 2) *
(-D + 4)) * G(m5) * G(m1) * G(m2) * G(m3) +
(2 * D - 4) * G(m5) * G(m3) * G(m2) * G(m1)))
t = -G(m0) * G(m1) * G(m2) * G(m3) * G(-m0) * G(m4)
st = tfunc(t)
assert _is_tensor_eq(st, ((D - 4) * G(m1) * G(m2) * G(m3) * G(m4) +
2 * G(m3) * G(m2) * G(m1) * G(m4)))
t = G(-m5) * G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(-m0) * G(m5)
st = tfunc(t)
result1 = ((-D + 4)**2 + 4)*G(m1)*G(m2)*G(m3)*G(m4) +\
(4*D - 16)*G(m3)*G(m2)*G(m1)*G(m4) + (4*D - 16)*G(m4)*G(m1)*G(m2)*G(m3)\
+ 4*G(m2)*G(m1)*G(m4)*G(m3) + 4*G(m3)*G(m4)*G(m1)*G(m2) +\
4*G(m4)*G(m3)*G(m2)*G(m1)
# Kahane's algorithm yields this result, which is equivalent to `result1`
# in four dimensions, but is not automatically recognized as equal:
result2 = 8 * G(m1) * G(m2) * G(m3) * G(m4) + 8 * G(m4) * G(m3) * G(
m2) * G(m1)
if D == 4:
assert _is_tensor_eq(st, (result1)) or _is_tensor_eq(st, (result2))
else:
assert _is_tensor_eq(st, (result1))
# and a few very simple cases, with no contracted indices:
t = G(m0)
st = tfunc(t)
assert _is_tensor_eq(st, t)
t = -7 * G(m0)
st = tfunc(t)
assert _is_tensor_eq(st, t)
t = 224 * G(m0) * G(m1) * G(-m2) * G(m3)
st = tfunc(t)
assert _is_tensor_eq(st, t)
def test_kahane_simplify1():
i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15 = tensor_indices('i0:16', LorentzIndex)
mu, nu, rho, sigma = tensor_indices("mu, nu, rho, sigma", LorentzIndex)
D = 4
t = G(i0)*G(i1)
r = kahane_simplify(t)
assert r.equals(t)
t = G(i0)*G(i1)*G(-i0)
r = kahane_simplify(t)
assert r.equals(-2*G(i1))
t = G(i0)*G(i1)*G(-i0)
r = kahane_simplify(t)
assert r.equals(-2*G(i1))
t = G(i0)*G(i1)
r = kahane_simplify(t)
assert r.equals(t)
t = G(i0)*G(i1)
r = kahane_simplify(t)
assert r.equals(t)
t = G(i0)*G(-i0)
r = kahane_simplify(t)
assert r.equals(4*eye(4))
t = G(i0)*G(-i0)
r = kahane_simplify(t)
assert r.equals(4*eye(4))
t = G(i0)*G(-i0)
r = kahane_simplify(t)
assert r.equals(4*eye(4))
t = G(i0)*G(i1)*G(-i0)
r = kahane_simplify(t)
assert r.equals(-2*G(i1))
t = G(i0)*G(i1)*G(-i0)*G(-i1)
r = kahane_simplify(t)
assert r.equals((2*D - D**2)*eye(4))
t = G(i0)*G(i1)*G(-i0)*G(-i1)
r = kahane_simplify(t)
assert r.equals((2*D - D**2)*eye(4))
t = G(i0)*G(-i0)*G(i1)*G(-i1)
r = kahane_simplify(t)
assert r.equals(16*eye(4))
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(D**2*eye(4))
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(D**2*eye(4))
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(D**2*eye(4))
t = (G(mu)*G(nu)*G(-rho)*G(-nu)*G(-mu)*G(rho))
r = kahane_simplify(t)
assert r.equals((4*D - 4*D**2 + D**3)*eye(4))
t = (G(-mu)*G(-nu)*G(-rho)*G(-sigma)*G(nu)*G(mu)*G(sigma)*G(rho))
r = kahane_simplify(t)
assert r.equals((-16*D + 24*D**2 - 8*D**3 + D**4)*eye(4))
t = (G(-mu)*G(nu)*G(-rho)*G(sigma)*G(rho)*G(-nu)*G(mu)*G(-sigma))
r = kahane_simplify(t)
assert r.equals((8*D - 12*D**2 + 6*D**3 - D**4)*eye(4))
# Expressions with free indices:
t = (G(mu)*G(nu)*G(rho)*G(sigma)*G(-mu))
r = kahane_simplify(t)
assert r.equals(-2*G(sigma)*G(rho)*G(nu))
t = (G(mu)*G(nu)*G(rho)*G(sigma)*G(-mu))
r = kahane_simplify(t)
assert r.equals(-2*G(sigma)*G(rho)*G(nu))
def test_gamma_matrix_trace():
gamma_trace = G.trace_tens
g = G.Lorentz.metric
m0, m1, m2, m3, m4, m5, m6 = tensor_indices('m0:7', G.Lorentz)
n0, n1, n2, n3, n4, n5 = tensor_indices('n0:6', G.Lorentz)
# working in D=4 dimensions
D = 4
# traces of odd number of gamma matrices are zero:
t = G(m0)
t1 = gamma_trace(t)
assert t1 == 0
t = G(m0) * G(m1) * G(m2)
t1 = gamma_trace(t)
assert t1 == 0
t = G(m0) * G(m1) * G(-m0)
t1 = gamma_trace(t)
assert t1 == 0
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4)
t1 = gamma_trace(t)
assert t1 == 0
# traces without internal contractions:
t = G(m0) * G(m1)
t1 = gamma_trace(t)
assert t1 == 4 * g(m0, m1)
t = G(m0) * G(m1) * G(m2) * G(m3)
t1 = gamma_trace(t)
t2 = -4 * g(m0, m2) * g(m1, m3) + 4 * g(m0, m1) * g(m2, m3) + 4 * g(
m0, m3) * g(m1, m2)
st2 = str(t2)
assert t1 == t2
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(m5)
t1 = gamma_trace(t)
t2 = t1 * g(-m0, -m5)
t2 = t2.contract_metric(g)
assert t2 == D * gamma_trace(G(m1) * G(m2) * G(m3) * G(m4))
# traces of expressions with internal contractions:
t = G(m0) * G(-m0)
t1 = gamma_trace(t)
assert t1 == 4 * D
t = G(m0) * G(m1) * G(-m0) * G(-m1)
t1 = gamma_trace(t)
assert t1 == 8 * D - 4 * D**2
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(-m0)
t1 = gamma_trace(t)
t2 = (-4*D)*g(m1, m3)*g(m2, m4) + (4*D)*g(m1, m2)*g(m3, m4) + \
(4*D)*g(m1, m4)*g(m2, m3)
assert t1 == t2
t = G(-m5) * G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(-m0) * G(m5)
t1 = gamma_trace(t)
t2 = (32*D + 4*(-D + 4)**2 - 64)*(g(m1, m2)*g(m3, m4) - \
g(m1, m3)*g(m2, m4) + g(m1, m4)*g(m2, m3))
assert t1 == t2
t = G(m0) * G(m1) * G(-m0) * G(m3)
t1 = gamma_trace(t)
assert t1 == (-4 * D + 8) * g(m1, m3)
# p, q = S1('p,q')
# ps = p(m0)*G(-m0)
# qs = q(m0)*G(-m0)
# t = ps*qs*ps*qs
# t1 = gamma_trace(t)
# assert t1 == 8*p(m0)*q(-m0)*p(m1)*q(-m1) - 4*p(m0)*p(-m0)*q(m1)*q(-m1)
t = G(m0) * G(m1) * G(m2) * G(m3) * G(m4) * G(m5) * G(-m0) * G(-m1) * G(
-m2) * G(-m3) * G(-m4) * G(-m5)
t1 = gamma_trace(t)
assert t1 == -4 * D**6 + 120 * D**5 - 1040 * D**4 + 3360 * D**3 - 4480 * D**2 + 2048 * D
t = G(m0) * G(m1) * G(n1) * G(m2) * G(n2) * G(m3) * G(m4) * G(-n2) * G(
-n1) * G(-m0) * G(-m1) * G(-m2) * G(-m3) * G(-m4)
t1 = gamma_trace(t)
tresu = -7168 * D + 16768 * D**2 - 14400 * D**3 + 5920 * D**4 - 1232 * D**5 + 120 * D**6 - 4 * D**7
assert t1 == tresu
# checked with Mathematica
# In[1]:= <<Tracer.m
# In[2]:= Spur[l];
# In[3]:= GammaTrace[l, {m0},{m1},{n1},{m2},{n2},{m3},{m4},{n3},{n4},{m0},{m1},{m2},{m3},{m4}]
t = G(m0) * G(m1) * G(n1) * G(m2) * G(n2) * G(m3) * G(m4) * G(n3) * G(
n4) * G(-m0) * G(-m1) * G(-m2) * G(-m3) * G(-m4)
t1 = gamma_trace(t)
# t1 = t1.expand_coeff()
c1 = -4 * D**5 + 120 * D**4 - 1200 * D**3 + 5280 * D**2 - 10560 * D + 7808
c2 = -4 * D**5 + 88 * D**4 - 560 * D**3 + 1440 * D**2 - 1600 * D + 640
assert t1 == c1*g(n1, n4)*g(n2, n3) + c2*g(n1, n2)*g(n3, n4) + \
(-c1)*g(n1, n3)*g(n2, n4)
import sympy as sp
from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex
from sympy.tensor.tensor import TensorIndex, tensor_indices, TensorIndexType
eps = LorentzIndex.epsilon
i0,i1,i2,i3,i4,i5 = tensor_indices('i_0:6', LorentzIndex)
G5 = 1j/ sp.factorial(4) * eps(-i0, -i1, -i2, -i3) * G(i0) * G(i1) * G(i2) * G(i3)
print(G5)
