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def test01():
__cdbkernel__=create_scope()
{m,n,p}::Indices(spacetime, position=fixed);
{a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed);
\sigma^{p}::ImplicitIndex(\sigma^{p a}_{b});
\psi::ImplicitIndex(\psi_{a});
\chi::ImplicitIndex(\chi^{a});
ex:= \psi \sigma^{m} \sigma^{n} \chi \lambda^{a} + 1/2 \lambda_{b} \chi^{b} T^{m n}\chi^{a};
explicit_indices(_)
tst:= \psi_{c} \sigma^{m c}_{d} \sigma^{n d}_{e} \chi^{e} \lambda^{a} + 1/2 \lambda_{b} \chi^{b} T^{m n}\chi^{a} - @(ex);
assert(tst==0)
print("Test 01a passed")
ex:= \psi \sigma^{m} \chi;
explicit_indices(_)
tst:= \psi_{a} \sigma^{m a}_{b} \chi^{b} - @(ex);
assert(tst==0)
print("Test 01b passed")
test01()
def test02():
__cdbkernel__=create_scope()
{m,n,p}::Indices(spacetime, position=fixed);
{a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed);
\sigma^{p}::ImplicitIndex(\sigma^{p a}_{b});
\psi::ImplicitIndex(\psi_{a});
\chi::ImplicitIndex(\chi^{a});
\partial{#}::PartialDerivative;
ex:= \psi \sigma^{m} \sigma^{n} \partial_{n}{\chi} \lambda^{a} + \lambda_{b} \chi^{b} T^{m}\chi^{a};
explicit_indices(_)
tst:= \psi_{c} \sigma^{m c}_{d} \sigma^{n d}_{e} \partial_{n}{\chi^{e}} \lambda^{a} + \lambda_{b} \chi^{b} T^{m}\chi^{a} - @(ex);
assert(tst==0)
print("Test 02 passed")
test02()
def test03():
__cdbkernel__=create_scope()
{m,n,p}::Indices(spacetime, position=fixed);
{a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed);
\sigma^{p}::ImplicitIndex(\sigma^{p a}_{b});
\psi::ImplicitIndex(\psi_{a});
\chi::ImplicitIndex(\chi^{a});
ex:= 2 \sigma^{m} \chi \psi \sigma^{n} \chi;
explicit_indices(_)
tst:= 2 \sigma^{m a}_{b} \chi^{b} \psi_{c} \sigma^{n c}_{d} \chi^{d} - @(ex);
assert(tst==0)
print("Test 03 passed")
test03()
def test04():
__cdbkernel__=create_scope()
{m,n,p}::Indices(spacetime, position=fixed);
{a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed);
\sigma^{p}::ImplicitIndex(\sigma^{p}_{a b});
\tau^{p}::ImplicitIndex(\tau^{p a b});
Tr{#}::Trace(indices=spinor);
ex:= Tr( \sigma^{m} \tau^{n} );
explicit_indices(_)
tst:= \sigma^{m}_{a b} \tau^{n b a} - @(ex);
assert(tst==0)
print("Test 04 passed")
test04()
def test05():
__cdbkernel__=create_scope()
{m,n,p}::Indices(spacetime, position=fixed);
{a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed);
\sigma^{p}::ImplicitIndex(\sigma^{p}_{a b});
\tau^{p}::ImplicitIndex(\tau^{p a b});
Tr{#}::Trace(indices=spinor);
ex:= 1/3 Tr( \sigma^{m} \tau^{n} + \tau^{n} \sigma^{m} );
explicit_indices(_)
tst:= 1/3 \sigma^{m}_{a b} \tau^{n b a} + 1/3 \tau^{n a b} \sigma^{m}_{b a} - @(ex);
assert(tst==0)
print("Test 05 passed")
test05()
# A more complicated example for the pattern matcher,
# matching indices to index values, both symbolically
# and numerically.
def test06():
__cdbkernel__=create_scope()
{r,t}::Coordinate;
{\mu,\nu}::Indices(spacetime, values={r,t});
{a,b,c,d}::Indices(group, values={0,1});
\partial{#}::PartialDerivative;
ex:= \partial_{\mu}(A_{a b}) \partial_{\mu}(A_{b a});
A_{a b}::Depends(r,t);
{q1,q2}::Depends(r,t);
rl:= { A_{0 0} = q1, A_{1 1} = q2 };
evaluate(ex, rl, simplify=False)
tst:= \partial_{t}{q1} \partial_{t}{q1}
+\partial_{r}{q1} \partial_{r}{q1}
+\partial_{t}{q2} \partial_{t}{q2}
+\partial_{r}{q2} \partial_{r}{q2}
- @(ex);
assert(tst==0)
print("Test 06 passed")
test06()
def test07():
__cdbkernel__=create_scope()
{r,t,v,w}::Coordinate;
{\mu,\nu}::Indices(spacetime, values={r,t});
{a,b,c,d}::Indices(group, values={v,w});
\partial{#}::PartialDerivative;
ex:= \partial_{\mu}(A_{a b}) \partial_{\mu}(A_{b a});
A_{a b}::Depends(r,t);
evaluate(ex, simplify=False)
tst:=\partial_{t}(A_{w w}) \partial_{t}(A_{w w}) + \partial_{r}(A_{w w}) \partial_{r}(A_{w w}) + \partial_{t}(A_{v w}) \partial_{t}(A_{w v}) + \partial_{r}(A_{v w}) \partial_{r}(A_{w v}) + \partial_{t}(A_{w v}) \partial_{t}(A_{v w}) + \partial_{r}(A_{w v}) \partial_{r}(A_{v w}) + \partial_{t}(A_{v v}) \partial_{t}(A_{v v}) + \partial_{r}(A_{v v}) \partial_{r}(A_{v v}) - @(ex);
assert(tst==0)
print("Test 07 passed")
test07()
def test08():
__cdbkernel__=create_scope()
{a,b,c,d}::Indices(vector).
tr{#}::Trace(indices=vector).
A::ImplicitIndex(A_{a b}).
B::ImplicitIndex(B_{a b}).
C::ImplicitIndex(C_{a b}).
D::ImplicitIndex(D_{a b}).
ex:=tr(c A B) tr(C D);
explicit_indices(_);
tst:= c A_{a b} B_{b a} C_{c d} D_{d c} - @(ex);
print("Test 08 passed")
test08()
# The original question:
# Tr{#}::Trace;
# {\mu,\nu}::Indices(spacetime, values={0,1,2,3});
# #, position=fixed);
# {r,t}::Coordinate;
# {a,b,c,d}::Indices(group, values={r,t});
# A::ImplicitIndex(A_{a b});
# \partial{#}::PartialDerivative;
# A::Depends(\partial{#});
# ex:= Tr( \partial_{\mu}{A} \partial^{\mu}{A} );
# explicit_indices(_);
# A_{a b}::Depends(r,t);
# #\partial{#});
# #{q1, q2}::Depends(\partial{#});
# # \mu
# rl:= { A_{r r} = q1, A_{t t} = q2 };
# evaluate(ex, simplify=False);
# evaluate(ex, rl, simplify=False);
# tst:= \partial_{\mu}{q1} \partial^{\mu}{q1} + \partial_{\mu}{q2} \partial^{\mu}{q2} - @(ex);
# assert(tst==0)
# print("Test 06 passed")
#ex:= \chi \psi \sigma^{n} \chi;
#explicit_indices(_);
# Normally index contraction aims for adjacent indices.
# With 'eager=True', contractions are generated as much
# as possible, even if that does not lead to adjacent indices.
#
# {m,n,p}::Indices(spacetime, position=fixed);
# {a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed);
# \sigma^{p}::ImplicitIndex(\sigma^{p}_{a b});
# \tau^{p}::ImplicitIndex(\tau^{p a b});
# \theta::ImplicitIndex(\theta^{a});
#
# ex := \sigma^{p} \theta \tau^{m};
# tst:= \sigma^{p}_{a b} \theta^{b} \tau^{m a c};
#
# The `combine` algorithm puts objects in the
# wrong order!
{A^{a \alpha}, B^{b}_{\alpha}}::NonCommuting;
A^{a}::ImplicitIndex(A^{a \alpha});
B^{b}::ImplicitIndex(B^{b}_{\alpha});
ex:=(A^a)^\alpha (B^b)_{\alpha};
print(tree(ex))
combine(ex);
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