|
|
from sympy.matrices.dense import eye, Matrixfrom sympy.tensor.tensor import tensor_indices, TensorHead, tensor_heads, \ TensExpr, canon_bpfrom sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex, \ kahane_simplify, gamma_trace, _simplify_single_line, simplify_gamma_expression
def _is_tensor_eq(arg1, arg2): arg1 = canon_bp(arg1) arg2 = canon_bp(arg2) if isinstance(arg1, TensExpr): return arg1.equals(arg2) elif isinstance(arg2, TensExpr): return arg2.equals(arg1) return arg1 == arg2
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", LorentzIndex) a1, a2, a3, a4, a5, a6 = tensor_indices("a1:7", LorentzIndex) mu11, mu12, mu21, mu31, mu32, mu41, mu51, mu52 = tensor_indices("mu11, mu12, mu21, mu31, mu32, mu41, mu51, mu52", LorentzIndex) mu61, mu71, mu72 = tensor_indices("mu61, mu71, mu72", LorentzIndex) m0, m1, m2, m3, m4, m5, m6 = tensor_indices("m0:7", 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 Lorentz-contracted expressions, these return scalars:
def add_delta(ne): return ne * eye(4) # 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_algorithm(): # Wrap this function to convert to and from TIDS:
def tfunc(e): return _simplify_single_line(e)
execute_gamma_simplify_tests_for_function(tfunc, D=4)
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_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_gamma_matrix_trace(): g = LorentzIndex.metric
m0, m1, m2, m3, m4, m5, m6 = tensor_indices('m0:7', LorentzIndex) n0, n1, n2, n3, n4, n5 = tensor_indices('n0:6', 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) 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 _is_tensor_eq(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 _is_tensor_eq(t1, 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 = tensor_heads('p,q', [LorentzIndex]) 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 _is_tensor_eq(r, -12*p2*pq*q2 + 16*pq*pq*pq) t = ps*qs*ps*qs*ps*qs*ps*qs r = gamma_trace(t) assert _is_tensor_eq(r, -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)
|
