717 lines
24 KiB
Python
717 lines
24 KiB
Python
"""
|
|
Module to handle gamma matrices expressed as tensor objects.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex
|
|
>>> from sympy.tensor.tensor import tensor_indices
|
|
>>> i = tensor_indices('i', LorentzIndex)
|
|
>>> G(i)
|
|
GammaMatrix(i)
|
|
|
|
Note that there is already an instance of GammaMatrixHead in four dimensions:
|
|
GammaMatrix, which is simply declare as
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix
|
|
>>> from sympy.tensor.tensor import tensor_indices
|
|
>>> i = tensor_indices('i', LorentzIndex)
|
|
>>> GammaMatrix(i)
|
|
GammaMatrix(i)
|
|
|
|
To access the metric tensor
|
|
|
|
>>> LorentzIndex.metric
|
|
metric(LorentzIndex,LorentzIndex)
|
|
|
|
"""
|
|
from sympy.core.mul import Mul
|
|
from sympy.core.singleton import S
|
|
from sympy.matrices.dense import eye
|
|
from sympy.matrices.expressions.trace import trace
|
|
from sympy.tensor.tensor import TensorIndexType, TensorIndex,\
|
|
TensMul, TensAdd, tensor_mul, Tensor, TensorHead, TensorSymmetry
|
|
|
|
|
|
# DiracSpinorIndex = TensorIndexType('DiracSpinorIndex', dim=4, dummy_name="S")
|
|
|
|
|
|
LorentzIndex = TensorIndexType('LorentzIndex', dim=4, dummy_name="L")
|
|
|
|
|
|
GammaMatrix = TensorHead("GammaMatrix", [LorentzIndex],
|
|
TensorSymmetry.no_symmetry(1), comm=None)
|
|
|
|
|
|
def extract_type_tens(expression, component):
|
|
"""
|
|
Extract from a ``TensExpr`` all tensors with `component`.
|
|
|
|
Returns two tensor expressions:
|
|
|
|
* the first contains all ``Tensor`` of having `component`.
|
|
* the second contains all remaining.
|
|
|
|
|
|
"""
|
|
if isinstance(expression, Tensor):
|
|
sp = [expression]
|
|
elif isinstance(expression, TensMul):
|
|
sp = expression.args
|
|
else:
|
|
raise ValueError('wrong type')
|
|
|
|
# Collect all gamma matrices of the same dimension
|
|
new_expr = S.One
|
|
residual_expr = S.One
|
|
for i in sp:
|
|
if isinstance(i, Tensor) and i.component == component:
|
|
new_expr *= i
|
|
else:
|
|
residual_expr *= i
|
|
return new_expr, residual_expr
|
|
|
|
|
|
def simplify_gamma_expression(expression):
|
|
extracted_expr, residual_expr = extract_type_tens(expression, GammaMatrix)
|
|
res_expr = _simplify_single_line(extracted_expr)
|
|
return res_expr * residual_expr
|
|
|
|
|
|
def simplify_gpgp(ex, sort=True):
|
|
"""
|
|
simplify products ``G(i)*p(-i)*G(j)*p(-j) -> p(i)*p(-i)``
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
|
|
LorentzIndex, simplify_gpgp
|
|
>>> from sympy.tensor.tensor import tensor_indices, tensor_heads
|
|
>>> p, q = tensor_heads('p, q', [LorentzIndex])
|
|
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', LorentzIndex)
|
|
>>> ps = p(i0)*G(-i0)
|
|
>>> qs = q(i0)*G(-i0)
|
|
>>> simplify_gpgp(ps*qs*qs)
|
|
GammaMatrix(-L_0)*p(L_0)*q(L_1)*q(-L_1)
|
|
"""
|
|
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
|
|
|
|
if sort:
|
|
ex = ex.sorted_components()
|
|
# this would be better off with pattern matching
|
|
while 1:
|
|
t = _simplify_gpgp(ex)
|
|
if t != ex:
|
|
ex = t
|
|
else:
|
|
return t
|
|
|
|
|
|
def gamma_trace(t):
|
|
"""
|
|
trace of a single line of gamma matrices
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
|
|
gamma_trace, LorentzIndex
|
|
>>> from sympy.tensor.tensor import tensor_indices, tensor_heads
|
|
>>> p, q = tensor_heads('p, q', [LorentzIndex])
|
|
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', LorentzIndex)
|
|
>>> ps = p(i0)*G(-i0)
|
|
>>> qs = q(i0)*G(-i0)
|
|
>>> gamma_trace(G(i0)*G(i1))
|
|
4*metric(i0, i1)
|
|
>>> gamma_trace(ps*ps) - 4*p(i0)*p(-i0)
|
|
0
|
|
>>> gamma_trace(ps*qs + ps*ps) - 4*p(i0)*p(-i0) - 4*p(i0)*q(-i0)
|
|
0
|
|
|
|
"""
|
|
if isinstance(t, TensAdd):
|
|
res = TensAdd(*[gamma_trace(x) for x in t.args])
|
|
return res
|
|
t = _simplify_single_line(t)
|
|
res = _trace_single_line(t)
|
|
return res
|
|
|
|
|
|
def _simplify_single_line(expression):
|
|
"""
|
|
Simplify single-line product of gamma matrices.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
|
|
LorentzIndex, _simplify_single_line
|
|
>>> from sympy.tensor.tensor import tensor_indices, TensorHead
|
|
>>> p = TensorHead('p', [LorentzIndex])
|
|
>>> i0,i1 = tensor_indices('i0:2', LorentzIndex)
|
|
>>> _simplify_single_line(G(i0)*G(i1)*p(-i1)*G(-i0)) + 2*G(i0)*p(-i0)
|
|
0
|
|
|
|
"""
|
|
t1, t2 = extract_type_tens(expression, GammaMatrix)
|
|
if t1 != 1:
|
|
t1 = kahane_simplify(t1)
|
|
res = t1*t2
|
|
return res
|
|
|
|
|
|
def _trace_single_line(t):
|
|
"""
|
|
Evaluate the trace of a single gamma matrix line inside a ``TensExpr``.
|
|
|
|
Notes
|
|
=====
|
|
|
|
If there are ``DiracSpinorIndex.auto_left`` and ``DiracSpinorIndex.auto_right``
|
|
indices trace over them; otherwise traces are not implied (explain)
|
|
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
|
|
LorentzIndex, _trace_single_line
|
|
>>> from sympy.tensor.tensor import tensor_indices, TensorHead
|
|
>>> p = TensorHead('p', [LorentzIndex])
|
|
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', LorentzIndex)
|
|
>>> _trace_single_line(G(i0)*G(i1))
|
|
4*metric(i0, i1)
|
|
>>> _trace_single_line(G(i0)*p(-i0)*G(i1)*p(-i1)) - 4*p(i0)*p(-i0)
|
|
0
|
|
|
|
"""
|
|
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
|
|
|
|
t = t.expand()
|
|
if isinstance(t, TensAdd):
|
|
a = [_trace_single_line1(x)*x.coeff for x in t.args]
|
|
return TensAdd(*a)
|
|
elif isinstance(t, (Tensor, TensMul)):
|
|
r = t.coeff*_trace_single_line1(t)
|
|
return r
|
|
else:
|
|
return trace(t)
|
|
|
|
|
|
def _gamma_trace1(*a):
|
|
gctr = 4 # FIXME specific for d=4
|
|
g = LorentzIndex.metric
|
|
if not a:
|
|
return gctr
|
|
n = len(a)
|
|
if n%2 == 1:
|
|
#return TensMul.from_data(S.Zero, [], [], [])
|
|
return S.Zero
|
|
if n == 2:
|
|
ind0 = a[0].get_indices()[0]
|
|
ind1 = a[1].get_indices()[0]
|
|
return gctr*g(ind0, ind1)
|
|
if n == 4:
|
|
ind0 = a[0].get_indices()[0]
|
|
ind1 = a[1].get_indices()[0]
|
|
ind2 = a[2].get_indices()[0]
|
|
ind3 = a[3].get_indices()[0]
|
|
|
|
return gctr*(g(ind0, ind1)*g(ind2, ind3) - \
|
|
g(ind0, ind2)*g(ind1, ind3) + g(ind0, ind3)*g(ind1, ind2))
|
|
|
|
|
|
def kahane_simplify(expression):
|
|
r"""
|
|
This function cancels contracted elements in a product of four
|
|
dimensional gamma matrices, resulting in an expression equal to the given
|
|
one, without the contracted gamma matrices.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
`expression` the tensor expression containing the gamma matrices to simplify.
|
|
|
|
Notes
|
|
=====
|
|
|
|
If spinor indices are given, the matrices must be given in
|
|
the order given in the product.
|
|
|
|
Algorithm
|
|
=========
|
|
|
|
The idea behind the algorithm is to use some well-known identities,
|
|
i.e., for contractions enclosing an even number of `\gamma` matrices
|
|
|
|
`\gamma^\mu \gamma_{a_1} \cdots \gamma_{a_{2N}} \gamma_\mu = 2 (\gamma_{a_{2N}} \gamma_{a_1} \cdots \gamma_{a_{2N-1}} + \gamma_{a_{2N-1}} \cdots \gamma_{a_1} \gamma_{a_{2N}} )`
|
|
|
|
for an odd number of `\gamma` matrices
|
|
|
|
`\gamma^\mu \gamma_{a_1} \cdots \gamma_{a_{2N+1}} \gamma_\mu = -2 \gamma_{a_{2N+1}} \gamma_{a_{2N}} \cdots \gamma_{a_{1}}`
|
|
|
|
Instead of repeatedly applying these identities to cancel out all contracted indices,
|
|
it is possible to recognize the links that would result from such an operation,
|
|
the problem is thus reduced to a simple rearrangement of free gamma matrices.
|
|
|
|
Examples
|
|
========
|
|
|
|
When using, always remember that the original expression coefficient
|
|
has to be handled separately
|
|
|
|
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex
|
|
>>> from sympy.physics.hep.gamma_matrices import kahane_simplify
|
|
>>> from sympy.tensor.tensor import tensor_indices
|
|
>>> i0, i1, i2 = tensor_indices('i0:3', LorentzIndex)
|
|
>>> ta = G(i0)*G(-i0)
|
|
>>> kahane_simplify(ta)
|
|
Matrix([
|
|
[4, 0, 0, 0],
|
|
[0, 4, 0, 0],
|
|
[0, 0, 4, 0],
|
|
[0, 0, 0, 4]])
|
|
>>> tb = G(i0)*G(i1)*G(-i0)
|
|
>>> kahane_simplify(tb)
|
|
-2*GammaMatrix(i1)
|
|
>>> t = G(i0)*G(-i0)
|
|
>>> kahane_simplify(t)
|
|
Matrix([
|
|
[4, 0, 0, 0],
|
|
[0, 4, 0, 0],
|
|
[0, 0, 4, 0],
|
|
[0, 0, 0, 4]])
|
|
>>> t = G(i0)*G(-i0)
|
|
>>> kahane_simplify(t)
|
|
Matrix([
|
|
[4, 0, 0, 0],
|
|
[0, 4, 0, 0],
|
|
[0, 0, 4, 0],
|
|
[0, 0, 0, 4]])
|
|
|
|
If there are no contractions, the same expression is returned
|
|
|
|
>>> tc = G(i0)*G(i1)
|
|
>>> kahane_simplify(tc)
|
|
GammaMatrix(i0)*GammaMatrix(i1)
|
|
|
|
References
|
|
==========
|
|
|
|
[1] Algorithm for Reducing Contracted Products of gamma Matrices,
|
|
Joseph Kahane, Journal of Mathematical Physics, Vol. 9, No. 10, October 1968.
|
|
"""
|
|
|
|
if isinstance(expression, Mul):
|
|
return expression
|
|
if isinstance(expression, TensAdd):
|
|
return TensAdd(*[kahane_simplify(arg) for arg in expression.args])
|
|
|
|
if isinstance(expression, Tensor):
|
|
return expression
|
|
|
|
assert isinstance(expression, TensMul)
|
|
|
|
gammas = expression.args
|
|
|
|
for gamma in gammas:
|
|
assert gamma.component == GammaMatrix
|
|
|
|
free = expression.free
|
|
# spinor_free = [_ for _ in expression.free_in_args if _[1] != 0]
|
|
|
|
# if len(spinor_free) == 2:
|
|
# spinor_free.sort(key=lambda x: x[2])
|
|
# assert spinor_free[0][1] == 1 and spinor_free[-1][1] == 2
|
|
# assert spinor_free[0][2] == 0
|
|
# elif spinor_free:
|
|
# raise ValueError('spinor indices do not match')
|
|
|
|
dum = []
|
|
for dum_pair in expression.dum:
|
|
if expression.index_types[dum_pair[0]] == LorentzIndex:
|
|
dum.append((dum_pair[0], dum_pair[1]))
|
|
|
|
dum = sorted(dum)
|
|
|
|
if len(dum) == 0: # or GammaMatrixHead:
|
|
# no contractions in `expression`, just return it.
|
|
return expression
|
|
|
|
# find the `first_dum_pos`, i.e. the position of the first contracted
|
|
# gamma matrix, Kahane's algorithm as described in his paper requires the
|
|
# gamma matrix expression to start with a contracted gamma matrix, this is
|
|
# a workaround which ignores possible initial free indices, and re-adds
|
|
# them later.
|
|
|
|
first_dum_pos = min(map(min, dum))
|
|
|
|
# for p1, p2, a1, a2 in expression.dum_in_args:
|
|
# if p1 != 0 or p2 != 0:
|
|
# # only Lorentz indices, skip Dirac indices:
|
|
# continue
|
|
# first_dum_pos = min(p1, p2)
|
|
# break
|
|
|
|
total_number = len(free) + len(dum)*2
|
|
number_of_contractions = len(dum)
|
|
|
|
free_pos = [None]*total_number
|
|
for i in free:
|
|
free_pos[i[1]] = i[0]
|
|
|
|
# `index_is_free` is a list of booleans, to identify index position
|
|
# and whether that index is free or dummy.
|
|
index_is_free = [False]*total_number
|
|
|
|
for i, indx in enumerate(free):
|
|
index_is_free[indx[1]] = True
|
|
|
|
# `links` is a dictionary containing the graph described in Kahane's paper,
|
|
# to every key correspond one or two values, representing the linked indices.
|
|
# All values in `links` are integers, negative numbers are used in the case
|
|
# where it is necessary to insert gamma matrices between free indices, in
|
|
# order to make Kahane's algorithm work (see paper).
|
|
links = {i: [] for i in range(first_dum_pos, total_number)}
|
|
|
|
# `cum_sign` is a step variable to mark the sign of every index, see paper.
|
|
cum_sign = -1
|
|
# `cum_sign_list` keeps storage for all `cum_sign` (every index).
|
|
cum_sign_list = [None]*total_number
|
|
block_free_count = 0
|
|
|
|
# multiply `resulting_coeff` by the coefficient parameter, the rest
|
|
# of the algorithm ignores a scalar coefficient.
|
|
resulting_coeff = S.One
|
|
|
|
# initialize a list of lists of indices. The outer list will contain all
|
|
# additive tensor expressions, while the inner list will contain the
|
|
# free indices (rearranged according to the algorithm).
|
|
resulting_indices = [[]]
|
|
|
|
# start to count the `connected_components`, which together with the number
|
|
# of contractions, determines a -1 or +1 factor to be multiplied.
|
|
connected_components = 1
|
|
|
|
# First loop: here we fill `cum_sign_list`, and draw the links
|
|
# among consecutive indices (they are stored in `links`). Links among
|
|
# non-consecutive indices will be drawn later.
|
|
for i, is_free in enumerate(index_is_free):
|
|
# if `expression` starts with free indices, they are ignored here;
|
|
# they are later added as they are to the beginning of all
|
|
# `resulting_indices` list of lists of indices.
|
|
if i < first_dum_pos:
|
|
continue
|
|
|
|
if is_free:
|
|
block_free_count += 1
|
|
# if previous index was free as well, draw an arch in `links`.
|
|
if block_free_count > 1:
|
|
links[i - 1].append(i)
|
|
links[i].append(i - 1)
|
|
else:
|
|
# Change the sign of the index (`cum_sign`) if the number of free
|
|
# indices preceding it is even.
|
|
cum_sign *= 1 if (block_free_count % 2) else -1
|
|
if block_free_count == 0 and i != first_dum_pos:
|
|
# check if there are two consecutive dummy indices:
|
|
# in this case create virtual indices with negative position,
|
|
# these "virtual" indices represent the insertion of two
|
|
# gamma^0 matrices to separate consecutive dummy indices, as
|
|
# Kahane's algorithm requires dummy indices to be separated by
|
|
# free indices. The product of two gamma^0 matrices is unity,
|
|
# so the new expression being examined is the same as the
|
|
# original one.
|
|
if cum_sign == -1:
|
|
links[-1-i] = [-1-i+1]
|
|
links[-1-i+1] = [-1-i]
|
|
if (i - cum_sign) in links:
|
|
if i != first_dum_pos:
|
|
links[i].append(i - cum_sign)
|
|
if block_free_count != 0:
|
|
if i - cum_sign < len(index_is_free):
|
|
if index_is_free[i - cum_sign]:
|
|
links[i - cum_sign].append(i)
|
|
block_free_count = 0
|
|
|
|
cum_sign_list[i] = cum_sign
|
|
|
|
# The previous loop has only created links between consecutive free indices,
|
|
# it is necessary to properly create links among dummy (contracted) indices,
|
|
# according to the rules described in Kahane's paper. There is only one exception
|
|
# to Kahane's rules: the negative indices, which handle the case of some
|
|
# consecutive free indices (Kahane's paper just describes dummy indices
|
|
# separated by free indices, hinting that free indices can be added without
|
|
# altering the expression result).
|
|
for i in dum:
|
|
# get the positions of the two contracted indices:
|
|
pos1 = i[0]
|
|
pos2 = i[1]
|
|
|
|
# create Kahane's upper links, i.e. the upper arcs between dummy
|
|
# (i.e. contracted) indices:
|
|
links[pos1].append(pos2)
|
|
links[pos2].append(pos1)
|
|
|
|
# create Kahane's lower links, this corresponds to the arcs below
|
|
# the line described in the paper:
|
|
|
|
# first we move `pos1` and `pos2` according to the sign of the indices:
|
|
linkpos1 = pos1 + cum_sign_list[pos1]
|
|
linkpos2 = pos2 + cum_sign_list[pos2]
|
|
|
|
# otherwise, perform some checks before creating the lower arcs:
|
|
|
|
# make sure we are not exceeding the total number of indices:
|
|
if linkpos1 >= total_number:
|
|
continue
|
|
if linkpos2 >= total_number:
|
|
continue
|
|
|
|
# make sure we are not below the first dummy index in `expression`:
|
|
if linkpos1 < first_dum_pos:
|
|
continue
|
|
if linkpos2 < first_dum_pos:
|
|
continue
|
|
|
|
# check if the previous loop created "virtual" indices between dummy
|
|
# indices, in such a case relink `linkpos1` and `linkpos2`:
|
|
if (-1-linkpos1) in links:
|
|
linkpos1 = -1-linkpos1
|
|
if (-1-linkpos2) in links:
|
|
linkpos2 = -1-linkpos2
|
|
|
|
# move only if not next to free index:
|
|
if linkpos1 >= 0 and not index_is_free[linkpos1]:
|
|
linkpos1 = pos1
|
|
|
|
if linkpos2 >=0 and not index_is_free[linkpos2]:
|
|
linkpos2 = pos2
|
|
|
|
# create the lower arcs:
|
|
if linkpos2 not in links[linkpos1]:
|
|
links[linkpos1].append(linkpos2)
|
|
if linkpos1 not in links[linkpos2]:
|
|
links[linkpos2].append(linkpos1)
|
|
|
|
# This loop starts from the `first_dum_pos` index (first dummy index)
|
|
# walks through the graph deleting the visited indices from `links`,
|
|
# it adds a gamma matrix for every free index in encounters, while it
|
|
# completely ignores dummy indices and virtual indices.
|
|
pointer = first_dum_pos
|
|
previous_pointer = 0
|
|
while True:
|
|
if pointer in links:
|
|
next_ones = links.pop(pointer)
|
|
else:
|
|
break
|
|
|
|
if previous_pointer in next_ones:
|
|
next_ones.remove(previous_pointer)
|
|
|
|
previous_pointer = pointer
|
|
|
|
if next_ones:
|
|
pointer = next_ones[0]
|
|
else:
|
|
break
|
|
|
|
if pointer == previous_pointer:
|
|
break
|
|
if pointer >=0 and free_pos[pointer] is not None:
|
|
for ri in resulting_indices:
|
|
ri.append(free_pos[pointer])
|
|
|
|
# The following loop removes the remaining connected components in `links`.
|
|
# If there are free indices inside a connected component, it gives a
|
|
# contribution to the resulting expression given by the factor
|
|
# `gamma_a gamma_b ... gamma_z + gamma_z ... gamma_b gamma_a`, in Kahanes's
|
|
# paper represented as {gamma_a, gamma_b, ... , gamma_z},
|
|
# virtual indices are ignored. The variable `connected_components` is
|
|
# increased by one for every connected component this loop encounters.
|
|
|
|
# If the connected component has virtual and dummy indices only
|
|
# (no free indices), it contributes to `resulting_indices` by a factor of two.
|
|
# The multiplication by two is a result of the
|
|
# factor {gamma^0, gamma^0} = 2 I, as it appears in Kahane's paper.
|
|
# Note: curly brackets are meant as in the paper, as a generalized
|
|
# multi-element anticommutator!
|
|
|
|
while links:
|
|
connected_components += 1
|
|
pointer = min(links.keys())
|
|
previous_pointer = pointer
|
|
# the inner loop erases the visited indices from `links`, and it adds
|
|
# all free indices to `prepend_indices` list, virtual indices are
|
|
# ignored.
|
|
prepend_indices = []
|
|
while True:
|
|
if pointer in links:
|
|
next_ones = links.pop(pointer)
|
|
else:
|
|
break
|
|
|
|
if previous_pointer in next_ones:
|
|
if len(next_ones) > 1:
|
|
next_ones.remove(previous_pointer)
|
|
|
|
previous_pointer = pointer
|
|
|
|
if next_ones:
|
|
pointer = next_ones[0]
|
|
|
|
if pointer >= first_dum_pos and free_pos[pointer] is not None:
|
|
prepend_indices.insert(0, free_pos[pointer])
|
|
# if `prepend_indices` is void, it means there are no free indices
|
|
# in the loop (and it can be shown that there must be a virtual index),
|
|
# loops of virtual indices only contribute by a factor of two:
|
|
if len(prepend_indices) == 0:
|
|
resulting_coeff *= 2
|
|
# otherwise, add the free indices in `prepend_indices` to
|
|
# the `resulting_indices`:
|
|
else:
|
|
expr1 = prepend_indices
|
|
expr2 = list(reversed(prepend_indices))
|
|
resulting_indices = [expri + ri for ri in resulting_indices for expri in (expr1, expr2)]
|
|
|
|
# sign correction, as described in Kahane's paper:
|
|
resulting_coeff *= -1 if (number_of_contractions - connected_components + 1) % 2 else 1
|
|
# power of two factor, as described in Kahane's paper:
|
|
resulting_coeff *= 2**(number_of_contractions)
|
|
|
|
# If `first_dum_pos` is not zero, it means that there are trailing free gamma
|
|
# matrices in front of `expression`, so multiply by them:
|
|
resulting_indices = [ free_pos[0:first_dum_pos] + ri for ri in resulting_indices ]
|
|
|
|
resulting_expr = S.Zero
|
|
for i in resulting_indices:
|
|
temp_expr = S.One
|
|
for j in i:
|
|
temp_expr *= GammaMatrix(j)
|
|
resulting_expr += temp_expr
|
|
|
|
t = resulting_coeff * resulting_expr
|
|
t1 = None
|
|
if isinstance(t, TensAdd):
|
|
t1 = t.args[0]
|
|
elif isinstance(t, TensMul):
|
|
t1 = t
|
|
if t1:
|
|
pass
|
|
else:
|
|
t = eye(4)*t
|
|
return t
|