Python tensor.tensorhead函数代码示例

def test_special_eq_ne():
    # test special equality cases:
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    a,b,d0,d1,i,j,k = tensor_indices('a,b,d0,d1,i,j,k', Lorentz)
    # A, B symmetric
    A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])
    p, q, r = tensorhead('p,q,r', [Lorentz], [[1]])

    t = 0*A(a, b)
    assert t == 0
    assert t == S.Zero

    assert p(i) != A(a, b)
    assert A(a, -a) != A(a, b)
    assert 0*(A(a, b) + B(a, b)) == 0
    assert 0*(A(a, b) + B(a, b)) == S.Zero

    assert 3*(A(a, b) - A(a, b)) == S.Zero

    assert p(i) + q(i) != A(a, b)
    assert p(i) + q(i) != A(a, b) + B(a, b)

    assert p(i) - p(i) == 0
    assert p(i) - p(i) == S.Zero

    assert A(a, b) == A(b, a)

def test_fun():
    D = Symbol('D')
    Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
    a,b,c,d,e = tensor_indices('a,b,c,d,e', Lorentz)
    g = Lorentz.metric

    p, q = tensorhead('p q', [Lorentz], [[1]])
    t = q(c)*p(a)*q(b) + g(a,b)*g(c,d)*q(-d)
    assert t(a,b,c) == t
    assert t - t(b,a,c) == q(c)*p(a)*q(b) - q(c)*p(b)*q(a)
    assert t(b,c,d) == q(d)*p(b)*q(c) + g(b,c)*g(d,e)*q(-e)
    t1 = t.fun_eval((a,b),(b,a))
    assert t1 == q(c)*p(b)*q(a) + g(a,b)*g(c,d)*q(-d)

    # check that g_{a b; c} = 0
    # example taken from  L. Brewin
    # "A brief introduction to Cadabra" arxiv:0903.2085
    # dg_{a b c} = \partial_{a} g_{b c} is symmetric in b, c
    dg = tensorhead('dg', [Lorentz]*3, [[1], [1]*2])
    # gamma^a_{b c} is the Christoffel symbol
    gamma = S.Half*g(a,d)*(dg(-b,-d,-c) + dg(-c,-b,-d) - dg(-d,-b,-c))
    # t = g_{a b; c}
    t = dg(-c,-a,-b) - g(-a,-d)*gamma(d,-b,-c) - g(-b,-d)*gamma(d,-a,-c)
    t = t.contract_metric(g, True)
    assert t == 0
    t = q(c)*p(a)*q(b)
    assert t(b,c,d) == q(d)*p(b)*q(c)

def test_TensorManager():
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    LorentzH = TensorIndexType('LorentzH', dummy_fmt='LH')
    i, j = tensor_indices('i,j', Lorentz)
    ih, jh = tensor_indices('ih,jh', LorentzH)
    p, q = tensorhead('p q', [Lorentz], [[1]])
    ph, qh = tensorhead('ph qh', [LorentzH], [[1]])

    Gsymbol = Symbol('Gsymbol')
    GHsymbol = Symbol('GHsymbol')
    TensorManager.set_comm(Gsymbol, GHsymbol, 0)
    G = tensorhead('G', [Lorentz], [[1]], Gsymbol)
    assert TensorManager._comm_i2symbol[G.comm] == Gsymbol
    GH = tensorhead('GH', [LorentzH], [[1]], GHsymbol)
    ps = G(i)*p(-i)
    psh = GH(ih)*ph(-ih)
    t = ps + psh
    t1 = t*t
    assert t1 == ps*ps + 2*ps*psh + psh*psh
    qs = G(i)*q(-i)
    qsh = GH(ih)*qh(-ih)
    assert ps*qsh == qsh*ps
    assert ps*qs != qs*ps
    n = TensorManager.comm_symbols2i(Gsymbol)
    assert TensorManager.comm_i2symbol(n) == Gsymbol

    assert GHsymbol in TensorManager._comm_symbols2i
    raises(ValueError, lambda: TensorManager.set_comm(GHsymbol, 1, 2))
    TensorManager.set_comms((Gsymbol,GHsymbol,0),(Gsymbol,1,1))
    assert TensorManager.get_comm(n, 1) == TensorManager.get_comm(1, n) == 1
    TensorManager.clear()
    assert TensorManager.comm == [{0:0, 1:0, 2:0}, {0:0, 1:1, 2:None}, {0:0, 1:None}]
    assert GHsymbol not in TensorManager._comm_symbols2i
    nh = TensorManager.comm_symbols2i(GHsymbol)
    assert GHsymbol in TensorManager._comm_symbols2i

def test_add2():
    Lorentz = TensorIndexType("Lorentz", dummy_fmt="L")
    m, n, p, q = tensor_indices("m,n,p,q", Lorentz)
    R = tensorhead("R", [Lorentz] * 4, [[2, 2]])
    A = tensorhead("A", [Lorentz] * 3, [[3]])
    t1 = 2 * R(m, n, p, q) - R(m, q, n, p) + R(m, p, n, q)
    t2 = t1 * A(-n, -p, -q)
    assert t2 == 0
    t1 = S(2) / 3 * R(m, n, p, q) - S(1) / 3 * R(m, q, n, p) + S(1) / 3 * R(m, p, n, q)
    t2 = t1 * A(-n, -p, -q)
    assert t2 == 0
    t = A(m, -m, n) + A(n, p, -p)
    assert t == 0

def test_add2():
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    m, n, p, q = tensor_indices('m,n,p,q', Lorentz)
    R = tensorhead('R', [Lorentz]*4, [[2, 2]])
    A = tensorhead('A', [Lorentz]*3, [[3]])
    t1 = 2*R(m, n, p, q) - R(m, q, n, p) + R(m, p, n, q)
    t2 = t1*A(-n, -p, -q)
    assert t2 == 0
    t1 = S(2)/3*R(m,n,p,q) - S(1)/3*R(m,q,n,p) + S(1)/3*R(m,p,n,q)
    t2 = t1*A(-n, -p, -q)
    assert t2 == 0
    t = A(m, -m, n) + A(n, p, -p)
    assert t == 0

def test_tensor_element():
    L = TensorIndexType("L")
    i, j, k, l, m, n = tensor_indices("i j k l m n", L)

    A = tensorhead("A", [L, L], [[1], [1]])

    a = A(i, j)

    assert isinstance(TensorElement(a, {}), Tensor)
    assert isinstance(TensorElement(a, {k: 1}), Tensor)

    te1 = TensorElement(a, {Symbol("i"): 1})
    assert te1.free == [(j, 0)]
    assert te1.get_free_indices() == [j]
    assert te1.dum == []

    te2 = TensorElement(a, {i: 1})
    assert te2.free == [(j, 0)]
    assert te2.get_free_indices() == [j]
    assert te2.dum == []

    assert te1 == te2

    array = Array([[1, 2], [3, 4]])
    assert te1.replace_with_arrays({A(i, j): array}, [j]) == array[1, :]

def test_riemann_invariants():
    Lorentz = TensorIndexType("Lorentz", dummy_fmt="L")
    d0, d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11 = tensor_indices("d0:12", Lorentz)
    # R^{d0 d1}_{d1 d0}; ord = [d0,-d0,d1,-d1]
    # T_c = -R^{d0 d1}_{d0 d1}
    R = tensorhead("R", [Lorentz] * 4, [[2, 2]])
    t = R(d0, d1, -d1, -d0)
    tc = t.canon_bp()
    assert str(tc) == "-R(L_0, L_1, -L_0, -L_1)"

    # R_d11^d1_d0^d5 * R^{d6 d4 d0}_d5 * R_{d7 d2 d8 d9} *
    # R_{d10 d3 d6 d4} * R^{d2 d7 d11}_d1 * R^{d8 d9 d3 d10}
    # can = [0,2,4,6, 1,3,8,10, 5,7,12,14, 9,11,16,18, 13,15,20,22,
    #        17,19,21<F10,23, 24,25]
    # T_c = R^{d0 d1 d2 d3} * R_{d0 d1}^{d4 d5} * R_{d2 d3}^{d6 d7} *
    # R_{d4 d5}^{d8 d9} * R_{d6 d7}^{d10 d11} * R_{d8 d9 d10 d11}

    t = (
        R(-d11, d1, -d0, d5)
        * R(d6, d4, d0, -d5)
        * R(-d7, -d2, -d8, -d9)
        * R(-d10, -d3, -d6, -d4)
        * R(d2, d7, d11, -d1)
        * R(d8, d9, d3, d10)
    )
    tc = t.canon_bp()
    assert (
        str(tc)
        == "R(L_0, L_1, L_2, L_3)*R(-L_0, -L_1, L_4, L_5)*R(-L_2, -L_3, L_6, L_7)*R(-L_4, -L_5, L_8, L_9)*R(-L_6, -L_7, L_10, L_11)*R(-L_8, -L_9, -L_10, -L_11)"
    )

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_canonicalize2():
    D = Symbol("D")
    Eucl = TensorIndexType("Eucl", metric=0, dim=D, dummy_fmt="E")
    i0, i1, i2, i3, i4, i5, i6, i7, i8, i9, i10, i11, i12, i13, i14 = tensor_indices("i0:15", Eucl)
    A = tensorhead("A", [Eucl] * 3, [[3]])

    # two examples from Cvitanovic, Group Theory page 59
    # of identities for antisymmetric tensors of rank 3
    # contracted according to the Kuratowski graph  eq.(6.59)
    t = A(i0, i1, i2) * A(-i1, i3, i4) * A(-i3, i7, i5) * A(-i2, -i5, i6) * A(-i4, -i6, i8)
    t1 = t.canon_bp()
    assert t1 == 0

    # eq.(6.60)
    # t = A(i0,i1,i2)*A(-i1,i3,i4)*A(-i2,i5,i6)*A(-i3,i7,i8)*A(-i6,-i7,i9)*
    #    A(-i8,i10,i13)*A(-i5,-i10,i11)*A(-i4,-i11,i12)*A(-i3,-i12,i14)
    t = (
        A(i0, i1, i2)
        * A(-i1, i3, i4)
        * A(-i2, i5, i6)
        * A(-i3, i7, i8)
        * A(-i6, -i7, i9)
        * A(-i8, i10, i13)
        * A(-i5, -i10, i11)
        * A(-i4, -i11, i12)
        * A(-i9, -i12, i14)
    )
    t1 = t.canon_bp()
    assert t1 == 0

def test_pprint():
    Lorentz = TensorIndexType('Lorentz')
    i0, i1, i2, i3, i4 = tensor_indices('i0:5', Lorentz)
    A = tensorhead('A', [Lorentz], [[1]])

    assert pretty(A) == "A(Lorentz)"
    assert pretty(A(i0)) == "A(i0)"

def test_TensorHead():
    assert TensAdd() == 0
    # simple example of algebraic expression
    Lorentz = TensorIndexType("Lorentz", dummy_fmt="L")
    a, b = tensor_indices("a,b", Lorentz)
    # A, B symmetric
    A = tensorhead("A", [Lorentz] * 2, [[1] * 2])
    assert A.rank == 2
    assert A.symmetry == tensorsymmetry([1] * 2)

def test_riemann_cyclic_replace():
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    m0, m1, m2, m3 = tensor_indices('m:4', Lorentz)
    symr = tensorsymmetry([2, 2])
    R = tensorhead('R', [Lorentz]*4, [[2, 2]])
    t = R(m0, m2, m1, m3)
    t1 = riemann_cyclic_replace(t)
    t1a = -S.One/3*R(m0, m3, m2, m1) + S.One/3*R(m0, m1, m2, m3) + Rational(2, 3)*R(m0, m2, m1, m3)
    assert t1 == t1a

def test_TensorHead():
    assert TensAdd() == 0
    # simple example of algebraic expression
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    a,b = tensor_indices('a,b', Lorentz)
    # A, B symmetric
    A = tensorhead('A', [Lorentz]*2, [[1]*2])
    assert A.rank == 2
    assert A.symmetry == tensorsymmetry([1]*2)

def test_TensorType():
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    sym = tensorsymmetry([1]*2)
    A = tensorhead('A', [Lorentz]*2, [[1]*2])
    assert A.typ == TensorType([Lorentz]*2, sym)
    assert A.types == [Lorentz]
    typ = TensorType([Lorentz]*2, sym)
    assert str(typ) == "TensorType(['Lorentz', 'Lorentz'])"
    raises(ValueError, lambda: typ(2))

def test_contract_metric1():
    D = Symbol('D')
    Lorentz = TensorIndexType('Lorentz', dim=D, dummy_fmt='L')
    a, b, c, d, e = tensor_indices('a,b,c,d,e', Lorentz)
    g = Lorentz.metric
    p = tensorhead('p', [Lorentz], [[1]])
    t = g(a, b)*p(-b)
    t1 = t.contract_metric(g)
    assert t1 == p(a)
    A, B = tensorhead('A,B', [Lorentz]*2, [[1]*2])

    # case with g with all free indices
    t1 = A(a,b)*B(-b,c)*g(d, e)
    t2 = t1.contract_metric(g)
    assert t1 == t2

    # case of g(d, -d)
    t1 = A(a,b)*B(-b,c)*g(-d, d)
    t2 = t1.contract_metric(g)
    assert t2 == D*A(a, d)*B(-d, c)

    # g with one free index
    t1 = A(a,b)*B(-b,-c)*g(c, d)
    t2 = t1.contract_metric(g)
    assert t2 == A(a, c)*B(-c, d)

    # g with both indices contracted with another tensor
    t1 = A(a,b)*B(-b,-c)*g(c, -a)
    t2 = t1.contract_metric(g)
    assert t2 == A(a, b)*B(-b, -a)

    t1 = A(a,b)*B(-b,-c)*g(c, d)*g(-a, -d)
    t2 = t1.contract_metric(g)
    t2 = t2.contract_metric(g)
    assert t2 == A(a,b)*B(-b,-a)

    t1 = A(a,b)*g(-a,-b)
    t2 = t1.contract_metric(g)
    assert t2 == A(a, -a)
    assert not t2.free
    Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
    a, b = tensor_indices('a,b', Lorentz)
    g = Lorentz.metric
    raises(ValueError, lambda: g(a, -a).contract_metric(g)) # no dim