def test_mul():
from sympy.abc import x
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
sym = tensorsymmetry([1] * 2)
t = TensMul.from_data(S.One, [], [], [])
assert str(t) == '1'
A, B = tensorhead('A B', [Lorentz] * 2, [[1] * 2])
t = (1 + x) * A(a, b)
assert str(t) == '(x + 1)*A(a, b)'
assert t.types == [Lorentz]
assert t.rank == 2
assert t.dum == []
assert t.coeff == 1 + x
assert sorted(t.free) == [(a, 0, 0), (b, 1, 0)]
assert t.components == [A]
t = A(-b, a) * B(-a, c) * A(-c, d)
t1 = tensor_mul(*t.split())
assert t == t(-b, d)
assert t == t1
assert tensor_mul(*[]) == TensMul.from_data(S.One, [], [], [])
t = TensMul.from_data(1, [], [], [])
zsym = tensorsymmetry()
typ = TensorType([], zsym)
C = typ('C')
assert str(C()) == 'C'
assert str(t) == '1'
assert t.split()[0] == t
raises(ValueError, lambda: TIDS.free_dum_from_indices(a, a))
raises(ValueError, lambda: TIDS.free_dum_from_indices(-a, -a))
raises(ValueError, lambda: A(a, b) * A(a, c))
t = A(a, b) * A(-a, c)
raises(ValueError, lambda: t(a, b, c))
def test_mul():
from sympy.abc import x
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b, c, d = tensor_indices('a,b,c,d', Lorentz)
sym = tensorsymmetry([1]*2)
t = TensMul.from_data(S.One, [], [], [])
assert str(t) == '1'
A, B = tensorhead('A B', [Lorentz]*2, [[1]*2])
t = (1 + x)*A(a, b)
assert str(t) == '(x + 1)*A(a, b)'
assert t.types == [Lorentz]
assert t.rank == 2
assert t.dum == []
assert t.coeff == 1 + x
assert sorted(t.free) == [(a, 0, 0), (b, 1, 0)]
assert t.components == [A]
t = A(-b, a)*B(-a, c)*A(-c, d)
t1 = tensor_mul(*t.split())
assert t == t(-b, d)
assert t == t1
assert tensor_mul(*[]) == TensMul.from_data(S.One, [], [], [])
t = TensMul.from_data(1, [], [], [])
zsym = tensorsymmetry()
typ = TensorType([], zsym)
C = typ('C')
assert str(C()) == 'C'
assert str(t) == '1'
assert t.split()[0] == t
raises(ValueError, lambda: TIDS.free_dum_from_indices(a, a))
raises(ValueError, lambda: TIDS.free_dum_from_indices(-a, -a))
raises(ValueError, lambda: A(a, b)*A(a, c))
t = A(a, b)*A(-a, c)
raises(ValueError, lambda: t(a, b, c))
def simplify_lines(ex):
"""
simplify a product of gamma matrices
Examples
========
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix, DiracSpinorIndex
>>> from sympy.tensor.tensor import tensor_indices
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', GammaMatrix.LorentzIndex)
>>> s0,s1,s2,s3,s4,s5,s6,s7 = tensor_indices('s0:8', DiracSpinorIndex)
>>> G = GammaMatrix
>>> t = G(i1,s1,-s2)*G(i4,s7,-s6)*G(i2,s2,-s3)*G(i3,s4,-s5)*G(i5,s6,-s7)
>>> G.simplify_lines(t)
4*gamma(i3, s4, -s5)*gamma(i1, s1, -S_0)*gamma(i2, S_0, -s3)*metric(i4, i5)
"""
lines, traces, rest = get_lines(ex, DiracSpinorIndex)
a = ex.split()
trest = tensor_mul(*[x for i, x in enumerate(a) if i in rest])
tlines = []
for line in lines:
first = a[line[0]]
last = a[line[-1]]
first = [x[0] for x in first.free if x[1] == 1][0]
last = [x[0] for x in last.free if x[1] == 2][0]
tx = tensor_mul(*[x for i, x in enumerate(a) if i in line])
tx1 = GammaMatrixHead._simplify_single_line(tx)
tlines.append(tx1)
traces = [GammaMatrix._trace_single_line(tensor_mul(*[x for i, x in enumerate(a) if i in line])) for line in traces]
res = tensor_mul(*([trest] + tlines + traces))
return res
def _trace_single_line1(t):
t = t.sorted_components()
components = t.components
ncomps = len(components)
g = LorentzIndex.metric
# gamma matirices are in a[i:j]
hit = 0
for i in range(ncomps):
if components[i] == GammaMatrix:
hit = 1
break
for j in range(i + hit, ncomps):
if components[j] != GammaMatrix:
break
else:
j = ncomps
numG = j - i
if numG == 0:
tcoeff = t.coeff
return t.nocoeff if tcoeff else t
if numG % 2 == 1:
return TensMul.from_data(S.Zero, [], [], [])
elif numG > 4:
# find the open matrix indices and connect them:
a = t.split()
ind1 = a[i].get_indices()[0]
ind2 = a[i + 1].get_indices()[0]
aa = a[:i] + a[i + 2:]
t1 = tensor_mul(*aa) * g(ind1, ind2)
t1 = t1.contract_metric(g)
args = [t1]
sign = 1
for k in range(i + 2, j):
sign = -sign
ind2 = a[k].get_indices()[0]
aa = a[:i] + a[i + 1:k] + a[k + 1:]
t2 = sign * tensor_mul(*aa) * g(ind1, ind2)
t2 = t2.contract_metric(g)
t2 = simplify_gpgp(t2, False)
args.append(t2)
t3 = TensAdd(*args)
t3 = _trace_single_line(t3)
return t3
else:
a = t.split()
t1 = _gamma_trace1(*a[i:j])
a2 = a[:i] + a[j:]
t2 = tensor_mul(*a2)
t3 = t1 * t2
if not t3:
return t3
t3 = t3.contract_metric(g)
return t3
def _trace_single_line1(t):
t = t.sorted_components()
components = t.components
ncomps = len(components)
g = LorentzIndex.metric
# gamma matirices are in a[i:j]
hit = 0
for i in range(ncomps):
if components[i] == GammaMatrix:
hit = 1
break
for j in range(i + hit, ncomps):
if components[j] != GammaMatrix:
break
else:
j = ncomps
numG = j - i
if numG == 0:
tcoeff = t.coeff
return t.nocoeff if tcoeff else t
if numG % 2 == 1:
return TensMul.from_data(S.Zero, [], [], [])
elif numG > 4:
# find the open matrix indices and connect them:
a = t.split()
ind1 = a[i].get_indices()[0]
ind2 = a[i + 1].get_indices()[0]
aa = a[:i] + a[i + 2:]
t1 = tensor_mul(*aa)*g(ind1, ind2)
t1 = t1.contract_metric(g)
args = [t1]
sign = 1
for k in range(i + 2, j):
sign = -sign
ind2 = a[k].get_indices()[0]
aa = a[:i] + a[i + 1:k] + a[k + 1:]
t2 = sign*tensor_mul(*aa)*g(ind1, ind2)
t2 = t2.contract_metric(g)
t2 = simplify_gpgp(t2, False)
args.append(t2)
t3 = TensAdd(*args)
t3 = _trace_single_line(t3)
return t3
else:
a = t.split()
t1 = _gamma_trace1(*a[i:j])
a2 = a[:i] + a[j:]
t2 = tensor_mul(*a2)
t3 = t1*t2
if not t3:
return t3
t3 = t3.contract_metric(g)
return t3
def _simplify_gpgp(ex):
components = ex.components
a = []
comp_map = []
for i, comp in enumerate(components):
comp_map.extend([i] * comp.rank)
dum = [(i[0], i[1], comp_map[i[0]], comp_map[i[1]]) for i in ex.dum]
for i in range(len(components)):
if components[i] != GammaMatrix:
continue
for dx in dum:
if dx[2] == i:
p_pos1 = dx[3]
elif dx[3] == i:
p_pos1 = dx[2]
else:
continue
comp1 = components[p_pos1]
if comp1.comm == 0 and comp1.rank == 1:
a.append((i, p_pos1))
if not a:
return ex
elim = set()
tv = []
hit = True
coeff = S.One
ta = None
while hit:
hit = False
for i, ai in enumerate(a[:-1]):
if ai[0] in elim:
continue
if ai[0] != a[i + 1][0] - 1:
continue
if components[ai[1]] != components[a[i + 1][1]]:
continue
elim.add(ai[0])
elim.add(ai[1])
elim.add(a[i + 1][0])
elim.add(a[i + 1][1])
if not ta:
ta = ex.split()
mu = TensorIndex('mu', LorentzIndex)
hit = True
if i == 0:
coeff = ex.coeff
tx = components[ai[1]](mu) * components[ai[1]](-mu)
if len(a) == 2:
tx *= 4 # eye(4)
tv.append(tx)
break
if tv:
a = [x for j, x in enumerate(ta) if j not in elim]
a.extend(tv)
t = tensor_mul(*a) * coeff
# t = t.replace(lambda x: x.is_Matrix, lambda x: 1)
return t
else:
return ex
def _simplify_gpgp(ex):
components = ex.components
a = []
comp_map = []
for i, comp in enumerate(components):
comp_map.extend([i]*comp.rank)
dum = [(i[0], i[1], comp_map[i[0]], comp_map[i[1]]) for i in ex.dum]
for i in range(len(components)):
if components[i] != GammaMatrix:
continue
for dx in dum:
if dx[2] == i:
p_pos1 = dx[3]
elif dx[3] == i:
p_pos1 = dx[2]
else:
continue
comp1 = components[p_pos1]
if comp1.comm == 0 and comp1.rank == 1:
a.append((i, p_pos1))
if not a:
return ex
elim = set()
tv = []
hit = True
coeff = S.One
ta = None
while hit:
hit = False
for i, ai in enumerate(a[:-1]):
if ai[0] in elim:
continue
if ai[0] != a[i + 1][0] - 1:
continue
if components[ai[1]] != components[a[i + 1][1]]:
continue
elim.add(ai[0])
elim.add(ai[1])
elim.add(a[i + 1][0])
elim.add(a[i + 1][1])
if not ta:
ta = ex.split()
mu = TensorIndex('mu', LorentzIndex)
hit = True
if i == 0:
coeff = ex.coeff
tx = components[ai[1]](mu)*components[ai[1]](-mu)
if len(a) == 2:
tx *= 4 # eye(4)
tv.append(tx)
break
if tv:
a = [x for j, x in enumerate(ta) if j not in elim]
a.extend(tv)
t = tensor_mul(*a)*coeff
# t = t.replace(lambda x: x.is_Matrix, lambda x: 1)
return t
else:
return ex
def _simplify_gpgp(ex):
tids = ex._tids
components = tids.components
a = []
for i in range(len(components)):
if not isinstance(components[i], GammaMatrixHead):
continue
dum = tids.dum
for dx in dum:
if dx[2] == i:
p_pos1 = dx[3]
elif dx[3] == i:
p_pos1 = dx[2]
else:
continue
comp1 = components[p_pos1]
if comp1.comm == 0 and comp1.rank == 1:
a.append((i, p_pos1))
if not a:
return ex
elim = set()
tv = []
hit = True
coeff = S.One
ta = None
while hit:
hit = False
for i, ai in enumerate(a[:-1]):
if ai[0] in elim:
continue
if ai[0] != a[i + 1][0] - 1:
continue
if components[ai[1]] != components[a[i + 1][1]]:
continue
elim.add(ai[0])
elim.add(ai[1])
elim.add(a[i + 1][0])
elim.add(a[i + 1][1])
if not ta:
ta = ex.split()
mu = TensorIndex('mu', GammaMatrix.LorentzIndex)
ind1 = ta[ai[0]].get_indices()[1]
ind2 = ta[ai[0] + 1].get_indices()[2]
hit = True
if i == 0:
coeff = ex.coeff
tx = components[ai[1]](mu)*components[ai[1]](-mu)
tv.append(tx*DiracSpinorIndex.delta(ind1, ind2))
break
if tv:
a = [x for j, x in enumerate(ta) if j not in elim]
a.extend(tv)
t = tensor_mul(*a)*coeff
t = t.contract_metric(DiracSpinorIndex.delta)
return t
else:
return ex
def even_trace_recursion(t):
assert isinstance(t, TensMul)
if t.rank == 0:
return t
elif t.rank == 2:
free1, free2 = t.free
return t.coeff * 4 * GammaMatrix.Lorentz.metric(free1[0], free2[0])
a = t.split()
for i in a:
assert isinstance(i.components[0], GammaMatrixHead)
ind1 = t.free[0][0]
r_list = []
metric_list = []
sign = 1
for k in range(1, len(a)):
ind2 = t.free[k][0]
aa1 = a[1:k] + a[k+1:]
r_list.append(sign*tensor_mul(*aa1))
metric_list.append(GammaMatrix.Lorentz.metric(ind1, ind2))
sign *= -1
p_list = [j*even_trace_recursion(i) for i, j in zip(r_list, metric_list)]
return t.coeff * TensAdd(*p_list)
def _trace_single_line1(t):
t = t.sorted_components()
components = t.components
ncomps = len(components)
g = self.LorentzIndex.metric
sg = DiracSpinorIndex.delta
# gamma matirices are in a[i:j]
hit = 0
for i in range(ncomps):
if isinstance(components[i], GammaMatrixHead):
hit = 1
break
for j in range(i + hit, ncomps):
if not isinstance(components[j], GammaMatrixHead):
break
else:
j = ncomps
numG = j - i
if numG == 0:
spinor_free = [_[0] for _ in t._tids.free if _[0].tensortype is DiracSpinorIndex]
tcoeff = t.coeff
if spinor_free == [DiracSpinorIndex.auto_left, -DiracSpinorIndex.auto_right]:
t = t*DiracSpinorIndex.delta(-DiracSpinorIndex.auto_left, DiracSpinorIndex.auto_right)
t = t.contract_metric(sg)
return t/tcoeff if tcoeff else t
else:
return t/tcoeff if tcoeff else t
if numG % 2 == 1:
return TensMul.from_data(S.Zero, [], [], [])
elif numG > 4:
t = t.substitute_indices((-DiracSpinorIndex.auto_right, -DiracSpinorIndex.auto_index), (DiracSpinorIndex.auto_left, DiracSpinorIndex.auto_index))
a = t.split()
ind1, lind1, rind1 = a[i].args[-1]
ind2, lind2, rind2 = a[i + 1].args[-1]
aa = a[:i] + a[i + 2:]
t1 = tensor_mul(*aa)*g(ind1, ind2)*sg(lind1, rind1)*sg(lind2, rind2)
t1 = t1.contract_metric(g)
t1 = t1.contract_metric(sg)
args = [t1]
sign = 1
for k in range(i + 2, j):
sign = -sign
ind2, lind2, rind2 = a[k].args[-1]
aa = a[:i] + a[i + 1:k] + a[k + 1:]
t2 = sign*tensor_mul(*aa)*g(ind1, ind2)*sg(lind1, rind1)*sg(lind2, rind2)
t2 = t2.contract_metric(g)
t2 = t2.contract_metric(sg)
t2 = GammaMatrixHead.simplify_gpgp(t2, False)
args.append(t2)
t3 = TensAdd(*args)
#aa = _tensorlist_contract_metric(aa, g(ind1, ind2))
#t3 = t3.canon_bp()
t3 = self._trace_single_line(t3)
return t3
else:
a = t.split()
if len(t.components) == 1:
if t.components[0] is DiracSpinorIndex.delta:
return 4 # FIXME only for D=4
t1 = self._gamma_trace1(*a[i:j])
a2 = a[:i] + a[j:]
t2 = tensor_mul(*a2)
t3 = t1*t2
if not t3:
return t3
t3 = t3.contract_metric(g)
return t3
