Source code for sympy.physics.secondquant

"""
Second quantization operators and states for bosons.

This follow the formulation of Fetter and Welecka, "Quantum Theory
of Many-Particle Systems."
"""
from __future__ import print_function, division

from collections import defaultdict

from sympy import (Add, Basic, cacheit, Dummy, Expr, Function, I,
                   KroneckerDelta, Mul, Pow, S, sqrt, Symbol, sympify, Tuple,
                   zeros)
from sympy.core.compatibility import reduce, xrange
from sympy.printing.str import StrPrinter

from sympy.physics.quantum.qexpr import split_commutative_parts
from sympy.core.compatibility import reduce
from sympy.utilities.iterables import has_dups
from sympy.utilities import default_sort_key

__all__ = [
    'Dagger',
    'KroneckerDelta',
    'BosonicOperator',
    'AnnihilateBoson',
    'CreateBoson',
    'AnnihilateFermion',
    'CreateFermion',
    'FockState',
    'FockStateBra',
    'FockStateKet',
    'FockStateBosonKet',
    'FockStateBosonBra',
    'BBra',
    'BKet',
    'FBra',
    'FKet',
    'F',
    'Fd',
    'B',
    'Bd',
    'apply_operators',
    'InnerProduct',
    'BosonicBasis',
    'VarBosonicBasis',
    'FixedBosonicBasis',
    'Commutator',
    'matrix_rep',
    'contraction',
    'wicks',
    'NO',
    'evaluate_deltas',
    'AntiSymmetricTensor',
    'substitute_dummies',
    'PermutationOperator',
    'simplify_index_permutations',
]


class SecondQuantizationError(Exception):
    pass


class AppliesOnlyToSymbolicIndex(SecondQuantizationError):
    pass


class ContractionAppliesOnlyToFermions(SecondQuantizationError):
    pass


class ViolationOfPauliPrinciple(SecondQuantizationError):
    pass


class SubstitutionOfAmbigousOperatorFailed(SecondQuantizationError):
    pass


class WicksTheoremDoesNotApply(SecondQuantizationError):
    pass


[docs]class Dagger(Expr): """ Hermitian conjugate of creation/annihilation operators. Examples ======== >>> from sympy import I >>> from sympy.physics.secondquant import Dagger, B, Bd >>> Dagger(2*I) -2*I >>> Dagger(B(0)) CreateBoson(0) >>> Dagger(Bd(0)) AnnihilateBoson(0) """ def __new__(cls, arg): arg = sympify(arg) r = cls.eval(arg) if isinstance(r, Basic): return r obj = Basic.__new__(cls, arg) return obj @classmethod
[docs] def eval(cls, arg): """ Evaluates the Dagger instance. Examples ======== >>> from sympy import I >>> from sympy.physics.secondquant import Dagger, B, Bd >>> Dagger(2*I) -2*I >>> Dagger(B(0)) CreateBoson(0) >>> Dagger(Bd(0)) AnnihilateBoson(0) The eval() method is called automatically. """ try: d = arg._dagger_() except AttributeError: if isinstance(arg, Basic): if arg.is_Add: return Add(*tuple(map(Dagger, arg.args))) if arg.is_Mul: return Mul(*tuple(map(Dagger, reversed(arg.args)))) if arg.is_Number: return arg if arg.is_Pow: return Pow(Dagger(arg.args[0]), arg.args[1]) if arg == I: return -arg else: return None else: return d
def _dagger_(self): return self.args[0]
class TensorSymbol(Expr): is_commutative = True
[docs]class AntiSymmetricTensor(TensorSymbol): """Stores upper and lower indices in separate Tuple's. Each group of indices is assumed to be antisymmetric. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import AntiSymmetricTensor >>> i, j = symbols('i j', below_fermi=True) >>> a, b = symbols('a b', above_fermi=True) >>> AntiSymmetricTensor('v', (a, i), (b, j)) AntiSymmetricTensor(v, (a, i), (b, j)) >>> AntiSymmetricTensor('v', (i, a), (b, j)) -AntiSymmetricTensor(v, (a, i), (b, j)) As you can see, the indices are automatically sorted to a canonical form. """ def __new__(cls, symbol, upper, lower): try: upper, signu = _sort_anticommuting_fermions( upper, key=cls._sortkey) lower, signl = _sort_anticommuting_fermions( lower, key=cls._sortkey) except ViolationOfPauliPrinciple: return S.Zero symbol = sympify(symbol) upper = Tuple(*upper) lower = Tuple(*lower) if (signu + signl) % 2: return -TensorSymbol.__new__(cls, symbol, upper, lower) else: return TensorSymbol.__new__(cls, symbol, upper, lower) @classmethod def _sortkey(cls, index): """Key for sorting of indices. particle < hole < general FIXME: This is a bottle-neck, can we do it faster? """ h = hash(index) if isinstance(index, Dummy): if index.assumptions0.get('above_fermi'): return (20, h) elif index.assumptions0.get('below_fermi'): return (21, h) else: return (22, h) if index.assumptions0.get('above_fermi'): return (10, h) elif index.assumptions0.get('below_fermi'): return (11, h) else: return (12, h) def _latex(self, printer): return "%s^{%s}_{%s}" % ( self.symbol, "".join([ i.name for i in self.args[1]]), "".join([ i.name for i in self.args[2]]) ) @property
[docs] def symbol(self): """ Returns the symbol of the tensor. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import AntiSymmetricTensor >>> i, j = symbols('i,j', below_fermi=True) >>> a, b = symbols('a,b', above_fermi=True) >>> AntiSymmetricTensor('v', (a, i), (b, j)) AntiSymmetricTensor(v, (a, i), (b, j)) >>> AntiSymmetricTensor('v', (a, i), (b, j)).symbol v """ return self.args[0]
@property
[docs] def upper(self): """ Returns the upper indices. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import AntiSymmetricTensor >>> i, j = symbols('i,j', below_fermi=True) >>> a, b = symbols('a,b', above_fermi=True) >>> AntiSymmetricTensor('v', (a, i), (b, j)) AntiSymmetricTensor(v, (a, i), (b, j)) >>> AntiSymmetricTensor('v', (a, i), (b, j)).upper (a, i) """ return self.args[1]
@property
[docs] def lower(self): """ Returns the lower indices. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import AntiSymmetricTensor >>> i, j = symbols('i,j', below_fermi=True) >>> a, b = symbols('a,b', above_fermi=True) >>> AntiSymmetricTensor('v', (a, i), (b, j)) AntiSymmetricTensor(v, (a, i), (b, j)) >>> AntiSymmetricTensor('v', (a, i), (b, j)).lower (b, j) """ return self.args[2]
def __str__(self): return "%s(%s,%s)" % self.args
[docs] def doit(self, **kw_args): """ Returns self. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import AntiSymmetricTensor >>> i, j = symbols('i,j', below_fermi=True) >>> a, b = symbols('a,b', above_fermi=True) >>> AntiSymmetricTensor('v', (a, i), (b, j)).doit() AntiSymmetricTensor(v, (a, i), (b, j)) """ return self
class SqOperator(Expr): """ Base class for Second Quantization operators. """ op_symbol = 'sq' is_commutative = False def __new__(cls, k): obj = Basic.__new__(cls, sympify(k)) return obj @property def state(self): """ Returns the state index related to this operator. >>> from sympy import Symbol >>> from sympy.physics.secondquant import F, Fd, B, Bd >>> p = Symbol('p') >>> F(p).state p >>> Fd(p).state p >>> B(p).state p >>> Bd(p).state p """ return self.args[0] @property def is_symbolic(self): """ Returns True if the state is a symbol (as opposed to a number). >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> p = Symbol('p') >>> F(p).is_symbolic True >>> F(1).is_symbolic False """ if self.state.is_Integer: return False else: return True def doit(self, **kw_args): """ FIXME: hack to prevent crash further up... """ return self def __repr__(self): return NotImplemented def __str__(self): return "%s(%r)" % (self.op_symbol, self.state) def apply_operator(self, state): """ Applies an operator to itself. """ raise NotImplementedError('implement apply_operator in a subclass') class BosonicOperator(SqOperator): pass class Annihilator(SqOperator): pass class Creator(SqOperator): pass
[docs]class AnnihilateBoson(BosonicOperator, Annihilator): """ Bosonic annihilation operator. Examples ======== >>> from sympy.physics.secondquant import B >>> from sympy.abc import x >>> B(x) AnnihilateBoson(x) """ op_symbol = 'b' def _dagger_(self): return CreateBoson(self.state)
[docs] def apply_operator(self, state): """ Apply state to self if self is not symbolic and state is a FockStateKet, else multiply self by state. Examples ======== >>> from sympy.physics.secondquant import B, BKet >>> from sympy.abc import x, y, n >>> B(x).apply_operator(y) y*AnnihilateBoson(x) >>> B(0).apply_operator(BKet((n,))) sqrt(n)*FockStateBosonKet((n - 1,)) """ if not self.is_symbolic and isinstance(state, FockStateKet): element = self.state amp = sqrt(state[element]) return amp*state.down(element) else: return Mul(self, state)
def __repr__(self): return "AnnihilateBoson(%s)" % self.state
[docs]class CreateBoson(BosonicOperator, Creator): """ Bosonic creation operator. """ op_symbol = 'b+' def _dagger_(self): return AnnihilateBoson(self.state)
[docs] def apply_operator(self, state): """ Apply state to self if self is not symbolic and state is a FockStateKet, else multiply self by state. Examples ======== >>> from sympy.physics.secondquant import B, Dagger, BKet >>> from sympy.abc import x, y, n >>> Dagger(B(x)).apply_operator(y) y*CreateBoson(x) >>> B(0).apply_operator(BKet((n,))) sqrt(n)*FockStateBosonKet((n - 1,)) """ if not self.is_symbolic and isinstance(state, FockStateKet): element = self.state amp = sqrt(state[element] + 1) return amp*state.up(element) else: return Mul(self, state)
def __repr__(self): return "CreateBoson(%s)" % self.state
B = AnnihilateBoson Bd = CreateBoson class FermionicOperator(SqOperator): @property def is_restricted(self): """ Is this FermionicOperator restricted with respect to fermi level? Return values: 1 : restricted to orbits above fermi 0 : no restriction -1 : restricted to orbits below fermi >>> from sympy import Symbol >>> from sympy.physics.secondquant import F, Fd >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_restricted 1 >>> Fd(a).is_restricted 1 >>> F(i).is_restricted -1 >>> Fd(i).is_restricted -1 >>> F(p).is_restricted 0 >>> Fd(p).is_restricted 0 """ ass = self.args[0].assumptions0 if ass.get("below_fermi"): return -1 if ass.get("above_fermi"): return 1 return 0 @property def is_above_fermi(self): """ Does the index of this FermionicOperator allow values above fermi? >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_above_fermi True >>> F(i).is_above_fermi False >>> F(p).is_above_fermi True The same applies to creation operators Fd """ return not self.args[0].assumptions0.get("below_fermi") @property def is_below_fermi(self): """ Does the index of this FermionicOperator allow values below fermi? >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_below_fermi False >>> F(i).is_below_fermi True >>> F(p).is_below_fermi True The same applies to creation operators Fd """ return not self.args[0].assumptions0.get("above_fermi") @property def is_only_below_fermi(self): """ Is the index of this FermionicOperator restricted to values below fermi? >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_only_below_fermi False >>> F(i).is_only_below_fermi True >>> F(p).is_only_below_fermi False The same applies to creation operators Fd """ return self.is_below_fermi and not self.is_above_fermi @property def is_only_above_fermi(self): """ Is the index of this FermionicOperator restricted to values above fermi? >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_only_above_fermi True >>> F(i).is_only_above_fermi False >>> F(p).is_only_above_fermi False The same applies to creation operators Fd """ return self.is_above_fermi and not self.is_below_fermi def _sortkey(self): h = hash(self) label = str(self.args[0]) if self.is_only_q_creator: return 1, label, h if self.is_only_q_annihilator: return 4, label, h if isinstance(self, Annihilator): return 3, label, h if isinstance(self, Creator): return 2, label, h
[docs]class AnnihilateFermion(FermionicOperator, Annihilator): """ Fermionic annihilation operator. """ op_symbol = 'f' def _dagger_(self): return CreateFermion(self.state)
[docs] def apply_operator(self, state): """ Apply state to self if self is not symbolic and state is a FockStateKet, else multiply self by state. Examples ======== >>> from sympy.physics.secondquant import B, Dagger, BKet >>> from sympy.abc import x, y, n >>> Dagger(B(x)).apply_operator(y) y*CreateBoson(x) >>> B(0).apply_operator(BKet((n,))) sqrt(n)*FockStateBosonKet((n - 1,)) """ if isinstance(state, FockStateFermionKet): element = self.state return state.down(element) elif isinstance(state, Mul): c_part, nc_part = state.args_cnc() if isinstance(nc_part[0], FockStateFermionKet): element = self.state return Mul(*(c_part + [nc_part[0].down(element)] + nc_part[1:])) else: return Mul(self, state) else: return Mul(self, state)
@property
[docs] def is_q_creator(self): """ Can we create a quasi-particle? (create hole or create particle) If so, would that be above or below the fermi surface? >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_q_creator 0 >>> F(i).is_q_creator -1 >>> F(p).is_q_creator -1 """ if self.is_below_fermi: return -1 return 0
@property
[docs] def is_q_annihilator(self): """ Can we destroy a quasi-particle? (annihilate hole or annihilate particle) If so, would that be above or below the fermi surface? >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=1) >>> i = Symbol('i', below_fermi=1) >>> p = Symbol('p') >>> F(a).is_q_annihilator 1 >>> F(i).is_q_annihilator 0 >>> F(p).is_q_annihilator 1 """ if self.is_above_fermi: return 1 return 0
@property
[docs] def is_only_q_creator(self): """ Always create a quasi-particle? (create hole or create particle) >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_only_q_creator False >>> F(i).is_only_q_creator True >>> F(p).is_only_q_creator False """ return self.is_only_below_fermi
@property
[docs] def is_only_q_annihilator(self): """ Always destroy a quasi-particle? (annihilate hole or annihilate particle) >>> from sympy import Symbol >>> from sympy.physics.secondquant import F >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> F(a).is_only_q_annihilator True >>> F(i).is_only_q_annihilator False >>> F(p).is_only_q_annihilator False """ return self.is_only_above_fermi
def __repr__(self): return "AnnihilateFermion(%s)" % self.state def _latex(self, printer): return "a_{%s}" % self.state.name
[docs]class CreateFermion(FermionicOperator, Creator): """ Fermionic creation operator. """ op_symbol = 'f+' def _dagger_(self): return AnnihilateFermion(self.state)
[docs] def apply_operator(self, state): """ Apply state to self if self is not symbolic and state is a FockStateKet, else multiply self by state. Examples ======== >>> from sympy.physics.secondquant import B, Dagger, BKet >>> from sympy.abc import x, y, n >>> Dagger(B(x)).apply_operator(y) y*CreateBoson(x) >>> B(0).apply_operator(BKet((n,))) sqrt(n)*FockStateBosonKet((n - 1,)) """ if isinstance(state, FockStateFermionKet): element = self.state return state.up(element) elif isinstance(state, Mul): c_part, nc_part = state.args_cnc() if isinstance(nc_part[0], FockStateFermionKet): element = self.state return Mul(*(c_part + [nc_part[0].up(element)] + nc_part[1:])) return Mul(self, state)
@property
[docs] def is_q_creator(self): """ Can we create a quasi-particle? (create hole or create particle) If so, would that be above or below the fermi surface? >>> from sympy import Symbol >>> from sympy.physics.secondquant import Fd >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> Fd(a).is_q_creator 1 >>> Fd(i).is_q_creator 0 >>> Fd(p).is_q_creator 1 """ if self.is_above_fermi: return 1 return 0
@property
[docs] def is_q_annihilator(self): """ Can we destroy a quasi-particle? (annihilate hole or annihilate particle) If so, would that be above or below the fermi surface? >>> from sympy import Symbol >>> from sympy.physics.secondquant import Fd >>> a = Symbol('a', above_fermi=1) >>> i = Symbol('i', below_fermi=1) >>> p = Symbol('p') >>> Fd(a).is_q_annihilator 0 >>> Fd(i).is_q_annihilator -1 >>> Fd(p).is_q_annihilator -1 """ if self.is_below_fermi: return -1 return 0
@property
[docs] def is_only_q_creator(self): """ Always create a quasi-particle? (create hole or create particle) >>> from sympy import Symbol >>> from sympy.physics.secondquant import Fd >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> Fd(a).is_only_q_creator True >>> Fd(i).is_only_q_creator False >>> Fd(p).is_only_q_creator False """ return self.is_only_above_fermi
@property
[docs] def is_only_q_annihilator(self): """ Always destroy a quasi-particle? (annihilate hole or annihilate particle) >>> from sympy import Symbol >>> from sympy.physics.secondquant import Fd >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> Fd(a).is_only_q_annihilator False >>> Fd(i).is_only_q_annihilator True >>> Fd(p).is_only_q_annihilator False """ return self.is_only_below_fermi
def __repr__(self): return "CreateFermion(%s)" % self.state def _latex(self, printer): return "a^\\dagger_{%s}" % self.state.name
Fd = CreateFermion F = AnnihilateFermion
[docs]class FockState(Expr): """ Many particle Fock state with a sequence of occupation numbers. Anywhere you can have a FockState, you can also have S.Zero. All code must check for this! Base class to represent FockStates. """ is_commutative = False def __new__(cls, occupations): """ occupations is a list with two possible meanings: - For bosons it is a list of occupation numbers. Element i is the number of particles in state i. - For fermions it is a list of occupied orbits. Element 0 is the state that was occupied first, element i is the i'th occupied state. """ occupations = list(map(sympify, occupations)) obj = Basic.__new__(cls, Tuple(*occupations)) return obj def __getitem__(self, i): i = int(i) return self.args[0][i] def __repr__(self): return ("FockState(%r)") % (self.args) def __str__(self): return "%s%r%s" % (self.lbracket, self._labels(), self.rbracket) def _labels(self): return self.args[0] def __len__(self): return len(self.args[0])
class BosonState(FockState): """ Base class for FockStateBoson(Ket/Bra). """ def up(self, i): """ Performs the action of a creation operator. Examples ======== >>> from sympy.physics.secondquant import BBra >>> b = BBra([1, 2]) >>> b FockStateBosonBra((1, 2)) >>> b.up(1) FockStateBosonBra((1, 3)) """ i = int(i) new_occs = list(self.args[0]) new_occs[i] = new_occs[i] + S.One return self.__class__(new_occs) def down(self, i): """ Performs the action of an annihilation operator. Examples ======== >>> from sympy.physics.secondquant import BBra >>> b = BBra([1, 2]) >>> b FockStateBosonBra((1, 2)) >>> b.down(1) FockStateBosonBra((1, 1)) """ i = int(i) new_occs = list(self.args[0]) if new_occs[i] == S.Zero: return S.Zero else: new_occs[i] = new_occs[i] - S.One return self.__class__(new_occs) class FermionState(FockState): """ Base class for FockStateFermion(Ket/Bra). """ fermi_level = 0 def __new__(cls, occupations, fermi_level=0): occupations = list(map(sympify, occupations)) if len(occupations) > 1: try: (occupations, sign) = _sort_anticommuting_fermions( occupations, key=hash) except ViolationOfPauliPrinciple: return S.Zero else: sign = 0 cls.fermi_level = fermi_level if cls._count_holes(occupations) > fermi_level: return S.Zero if sign % 2: return S.NegativeOne*FockState.__new__(cls, occupations) else: return FockState.__new__(cls, occupations) def up(self, i): """ Performs the action of a creation operator. If below fermi we try to remove a hole, if above fermi we try to create a particle. if general index p we return Kronecker(p,i)*self where i is a new symbol with restriction above or below. >>> from sympy import Symbol >>> from sympy.physics.secondquant import FKet >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> FKet([]).up(a) FockStateFermionKet((a,)) A creator acting on vacuum below fermi vanishes >>> FKet([]).up(i) 0 """ present = i in self.args[0] if self._only_above_fermi(i): if present: return S.Zero else: return self._add_orbit(i) elif self._only_below_fermi(i): if present: return self._remove_orbit(i) else: return S.Zero else: if present: hole = Dummy("i", below_fermi=True) return KroneckerDelta(i, hole)*self._remove_orbit(i) else: particle = Dummy("a", above_fermi=True) return KroneckerDelta(i, particle)*self._add_orbit(i) def down(self, i): """ Performs the action of an annihilation operator. If below fermi we try to create a hole, if above fermi we try to remove a particle. if general index p we return Kronecker(p,i)*self where i is a new symbol with restriction above or below. >>> from sympy import Symbol >>> from sympy.physics.secondquant import FKet >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') An annihilator acting on vacuum above fermi vanishes >>> FKet([]).down(a) 0 Also below fermi, it vanishes, unless we specify a fermi level > 0 >>> FKet([]).down(i) 0 >>> FKet([],4).down(i) FockStateFermionKet((i,)) """ present = i in self.args[0] if self._only_above_fermi(i): if present: return self._remove_orbit(i) else: return S.Zero elif self._only_below_fermi(i): if present: return S.Zero else: return self._add_orbit(i) else: if present: hole = Dummy("i", below_fermi=True) return KroneckerDelta(i, hole)*self._add_orbit(i) else: particle = Dummy("a", above_fermi=True) return KroneckerDelta(i, particle)*self._remove_orbit(i) @classmethod def _only_below_fermi(cls, i): """ Tests if given orbit is only below fermi surface. If nothing can be concluded we return a conservative False. """ if i.is_number: return i <= cls.fermi_level if i.assumptions0.get('below_fermi'): return True return False @classmethod def _only_above_fermi(cls, i): """ Tests if given orbit is only above fermi surface. If fermi level has not been set we return True. If nothing can be concluded we return a conservative False. """ if i.is_number: return i > cls.fermi_level if i.assumptions0.get('above_fermi'): return True return not cls.fermi_level def _remove_orbit(self, i): """ Removes particle/fills hole in orbit i. No input tests performed here. """ new_occs = list(self.args[0]) pos = new_occs.index(i) del new_occs[pos] if (pos) % 2: return S.NegativeOne*self.__class__(new_occs, self.fermi_level) else: return self.__class__(new_occs, self.fermi_level) def _add_orbit(self, i): """ Adds particle/creates hole in orbit i. No input tests performed here. """ return self.__class__((i,) + self.args[0], self.fermi_level) @classmethod def _count_holes(cls, list): """ returns number of identified hole states in list. """ return len([i for i in list if cls._only_below_fermi(i)]) def _negate_holes(self, list): return tuple([-i if i <= self.fermi_level else i for i in list]) def __repr__(self): if self.fermi_level: return "FockStateKet(%r, fermi_level=%s)" % (self.args[0], self.fermi_level) else: return "FockStateKet(%r)" % (self.args[0],) def _labels(self): return self._negate_holes(self.args[0])
[docs]class FockStateKet(FockState): """ Representation of a ket. """ lbracket = '|' rbracket = '>'
[docs]class FockStateBra(FockState): """ Representation of a bra. """ lbracket = '<' rbracket = '|' def __mul__(self, other): if isinstance(other, FockStateKet): return InnerProduct(self, other) else: return Expr.__mul__(self, other)
[docs]class FockStateBosonKet(BosonState, FockStateKet): """ Many particle Fock state with a sequence of occupation numbers. Occupation numbers can be any integer >= 0. Examples ======== >>> from sympy.physics.secondquant import BKet >>> BKet([1, 2]) FockStateBosonKet((1, 2)) """ def _dagger_(self): return FockStateBosonBra(*self.args)
[docs]class FockStateBosonBra(BosonState, FockStateBra): """ Describes a collection of BosonBra particles. Examples ======== >>> from sympy.physics.secondquant import BBra >>> BBra([1, 2]) FockStateBosonBra((1, 2)) """ def _dagger_(self): return FockStateBosonKet(*self.args)
class FockStateFermionKet(FermionState, FockStateKet): """ Many-particle Fock state with a sequence of occupied orbits. Each state can only have one particle, so we choose to store a list of occupied orbits rather than a tuple with occupation numbers (zeros and ones). states below fermi level are holes, and are represented by negative labels in the occupation list. For symbolic state labels, the fermi_level caps the number of allowed hole- states. Examples ======== >>> from sympy.physics.secondquant import FKet >>> FKet([1, 2]) #doctest: +SKIP FockStateFermionKet((1, 2)) """ def _dagger_(self): return FockStateFermionBra(*self.args) class FockStateFermionBra(FermionState, FockStateBra): """ See Also ======== FockStateFermionKet Examples ======== >>> from sympy.physics.secondquant import FBra >>> FBra([1, 2]) #doctest: +SKIP FockStateFermionBra((1, 2)) """ def _dagger_(self): return FockStateFermionKet(*self.args) BBra = FockStateBosonBra BKet = FockStateBosonKet FBra = FockStateFermionBra FKet = FockStateFermionKet def _apply_Mul(m): """ Take a Mul instance with operators and apply them to states. This method applies all operators with integer state labels to the actual states. For symbolic state labels, nothing is done. When inner products of FockStates are encountered (like <a|b>), they are converted to instances of InnerProduct. This does not currently work on double inner products like, <a|b><c|d>. If the argument is not a Mul, it is simply returned as is. """ if not isinstance(m, Mul): return m c_part, nc_part = m.args_cnc() n_nc = len(nc_part) if n_nc == 0 or n_nc == 1: return m else: last = nc_part[-1] next_to_last = nc_part[-2] if isinstance(last, FockStateKet): if isinstance(next_to_last, SqOperator): if next_to_last.is_symbolic: return m else: result = next_to_last.apply_operator(last) if result == 0: return S.Zero else: return _apply_Mul(Mul(*(c_part + nc_part[:-2] + [result]))) elif isinstance(next_to_last, Pow): if isinstance(next_to_last.base, SqOperator) and \ next_to_last.exp.is_Integer: if next_to_last.base.is_symbolic: return m else: result = last for i in range(next_to_last.exp): result = next_to_last.base.apply_operator(result) if result == 0: break if result == 0: return S.Zero else: return _apply_Mul(Mul(*(c_part + nc_part[:-2] + [result]))) else: return m elif isinstance(next_to_last, FockStateBra): result = InnerProduct(next_to_last, last) if result == 0: return S.Zero else: return _apply_Mul(Mul(*(c_part + nc_part[:-2] + [result]))) else: return m else: return m
[docs]def apply_operators(e): """ Take a sympy expression with operators and states and apply the operators. Examples ======== >>> from sympy.physics.secondquant import apply_operators >>> from sympy import sympify >>> apply_operators(sympify(3)+4) 7 """ e = e.expand() muls = e.atoms(Mul) subs_list = [(m, _apply_Mul(m)) for m in iter(muls)] return e.subs(subs_list)
[docs]class InnerProduct(Basic): """ An unevaluated inner product between a bra and ket. Currently this class just reduces things to a product of Kronecker Deltas. In the future, we could introduce abstract states like ``|a>`` and ``|b>``, and leave the inner product unevaluated as ``<a|b>``. """ is_commutative = True def __new__(cls, bra, ket): if not isinstance(bra, FockStateBra): raise TypeError("must be a bra") if not isinstance(ket, FockStateKet): raise TypeError("must be a key") return cls.eval(bra, ket) @classmethod def eval(cls, bra, ket): result = S.One for i, j in zip(bra.args[0], ket.args[0]): result *= KroneckerDelta(i, j) if result == 0: break return result @property
[docs] def bra(self): """Returns the bra part of the state""" return self.args[0]
@property
[docs] def ket(self): """Returns the ket part of the state""" return self.args[1]
def __repr__(self): sbra = repr(self.bra) sket = repr(self.ket) return "%s|%s" % (sbra[:-1], sket[1:]) def __str__(self): return self.__repr__()
[docs]def matrix_rep(op, basis): """ Find the representation of an operator in a basis. Examples ======== >>> from sympy.physics.secondquant import VarBosonicBasis, B, matrix_rep >>> b = VarBosonicBasis(5) >>> o = B(0) >>> matrix_rep(o, b) Matrix([ [0, 1, 0, 0, 0], [0, 0, sqrt(2), 0, 0], [0, 0, 0, sqrt(3), 0], [0, 0, 0, 0, 2], [0, 0, 0, 0, 0]]) """ a = zeros(len(basis)) for i in range(len(basis)): for j in range(len(basis)): a[i, j] = apply_operators(Dagger(basis[i])*op*basis[j]) return a
[docs]class BosonicBasis(object): """ Base class for a basis set of bosonic Fock states. """ pass
[docs]class VarBosonicBasis(object): """ A single state, variable particle number basis set. Examples ======== >>> from sympy.physics.secondquant import VarBosonicBasis >>> b = VarBosonicBasis(5) >>> b [FockState((0,)), FockState((1,)), FockState((2,)), FockState((3,)), FockState((4,))] """ def __init__(self, n_max): self.n_max = n_max self._build_states() def _build_states(self): self.basis = [] for i in range(self.n_max): self.basis.append(FockStateBosonKet([i])) self.n_basis = len(self.basis)
[docs] def index(self, state): """ Returns the index of state in basis. Examples ======== >>> from sympy.physics.secondquant import VarBosonicBasis >>> b = VarBosonicBasis(3) >>> state = b.state(1) >>> b [FockState((0,)), FockState((1,)), FockState((2,))] >>> state FockStateBosonKet((1,)) >>> b.index(state) 1 """ return self.basis.index(state)
[docs] def state(self, i): """ The state of a single basis. Examples ======== >>> from sympy.physics.secondquant import VarBosonicBasis >>> b = VarBosonicBasis(5) >>> b.state(3) FockStateBosonKet((3,)) """ return self.basis[i]
def __getitem__(self, i): return self.state(i) def __len__(self): return len(self.basis) def __repr__(self): return repr(self.basis)
[docs]class FixedBosonicBasis(BosonicBasis): """ Fixed particle number basis set. Examples ======== >>> from sympy.physics.secondquant import FixedBosonicBasis >>> b = FixedBosonicBasis(2, 2) >>> state = b.state(1) >>> b [FockState((2, 0)), FockState((1, 1)), FockState((0, 2))] >>> state FockStateBosonKet((1, 1)) >>> b.index(state) 1 """ def __init__(self, n_particles, n_levels): self.n_particles = n_particles self.n_levels = n_levels self._build_particle_locations() self._build_states() def _build_particle_locations(self): tup = ["i%i" % i for i in range(self.n_particles)] first_loop = "for i0 in range(%i)" % self.n_levels other_loops = '' for cur, prev in zip(tup[1:], tup): temp = "for %s in range(%s + 1) " % (cur, prev) other_loops = other_loops + temp tup_string = "(%s)" % ", ".join(tup) list_comp = "[%s %s %s]" % (tup_string, first_loop, other_loops) result = eval(list_comp) if self.n_particles == 1: result = [(item,) for item in result] self.particle_locations = result def _build_states(self): self.basis = [] for tuple_of_indices in self.particle_locations: occ_numbers = self.n_levels*[0] for level in tuple_of_indices: occ_numbers[level] += 1 self.basis.append(FockStateBosonKet(occ_numbers)) self.n_basis = len(self.basis)
[docs] def index(self, state): """Returns the index of state in basis. Examples ======== >>> from sympy.physics.secondquant import FixedBosonicBasis >>> b = FixedBosonicBasis(2, 3) >>> b.index(b.state(3)) 3 """ return self.basis.index(state)
[docs] def state(self, i): """Returns the state that lies at index i of the basis Examples ======== >>> from sympy.physics.secondquant import FixedBosonicBasis >>> b = FixedBosonicBasis(2, 3) >>> b.state(3) FockStateBosonKet((1, 0, 1)) """ return self.basis[i]
def __getitem__(self, i): return self.state(i) def __len__(self): return len(self.basis) def __repr__(self): return repr(self.basis) # def move(e, i, d): # """ # Takes the expression "e" and moves the operator at the position i by "d". # """ # if e.is_Mul: # if d == 1: # # e = a*b*c*d # a = Mul(*e.args[:i]) # b = e.args[i] # c = e.args[i+1] # d = Mul(*e.args[i+2:]) # if isinstance(b, Dagger) and not isinstance(c, Dagger): # i, j = b.args[0].args[0], c.args[0] # return a*c*b*d-a*KroneckerDelta(i, j)*d # elif not isinstance(b, Dagger) and isinstance(c, Dagger): # i, j = b.args[0], c.args[0].args[0] # return a*c*b*d-a*KroneckerDelta(i, j)*d # else: # return a*c*b*d # elif d == -1: # # e = a*b*c*d # a = Mul(*e.args[:i-1]) # b = e.args[i-1] # c = e.args[i] # d = Mul(*e.args[i+1:]) # if isinstance(b, Dagger) and not isinstance(c, Dagger): # i, j = b.args[0].args[0], c.args[0] # return a*c*b*d-a*KroneckerDelta(i, j)*d # elif not isinstance(b, Dagger) and isinstance(c, Dagger): # i, j = b.args[0], c.args[0].args[0] # return a*c*b*d-a*KroneckerDelta(i, j)*d # else: # return a*c*b*d # else: # if d > 1: # while d >= 1: # e = move(e, i, 1) # d -= 1 # i += 1 # return e # elif d < -1: # while d <= -1: # e = move(e, i, -1) # d += 1 # i -= 1 # return e # elif isinstance(e, Add): # a, b = e.as_two_terms() # return move(a, i, d) + move(b, i, d) # raise NotImplementedError()
[docs]class Commutator(Function): """ The Commutator: [A, B] = A*B - B*A The arguments are ordered according to .__cmp__() >>> from sympy import symbols >>> from sympy.physics.secondquant import Commutator >>> A, B = symbols('A,B', commutative=False) >>> Commutator(B, A) -Commutator(A, B) Evaluate the commutator with .doit() >>> comm = Commutator(A,B); comm Commutator(A, B) >>> comm.doit() A*B - B*A For two second quantization operators the commutator is evaluated immediately: >>> from sympy.physics.secondquant import Fd, F >>> a = symbols('a', above_fermi=True) >>> i = symbols('i', below_fermi=True) >>> p,q = symbols('p,q') >>> Commutator(Fd(a),Fd(i)) 2*NO(CreateFermion(a)*CreateFermion(i)) But for more complicated expressions, the evaluation is triggered by a call to .doit() >>> comm = Commutator(Fd(p)*Fd(q),F(i)); comm Commutator(CreateFermion(p)*CreateFermion(q), AnnihilateFermion(i)) >>> comm.doit(wicks=True) -KroneckerDelta(i, p)*CreateFermion(q) + KroneckerDelta(i, q)*CreateFermion(p) """ is_commutative = False @classmethod
[docs] def eval(cls, a, b): """ The Commutator [A,B] is on canonical form if A < B. Examples ======== >>> from sympy.physics.secondquant import Commutator, F, Fd >>> from sympy.abc import x >>> c1 = Commutator(F(x), Fd(x)) >>> c2 = Commutator(Fd(x), F(x)) >>> Commutator.eval(c1, c2) 0 """ if not (a and b): return S.Zero if a == b: return S.Zero if a.is_commutative or b.is_commutative: return S.Zero # # [A+B,C] -> [A,C] + [B,C] # a = a.expand() if isinstance(a, Add): return Add(*[cls(term, b) for term in a.args]) b = b.expand() if isinstance(b, Add): return Add(*[cls(a, term) for term in b.args]) # # [xA,yB] -> xy*[A,B] # ca, nca = a.args_cnc() cb, ncb = b.args_cnc() c_part = list(ca) + list(cb) if c_part: return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb))) # # single second quantization operators # if isinstance(a, BosonicOperator) and isinstance(b, BosonicOperator): if isinstance(b, CreateBoson) and isinstance(a, AnnihilateBoson): return KroneckerDelta(a.state, b.state) if isinstance(a, CreateBoson) and isinstance(b, AnnihilateBoson): return S.NegativeOne*KroneckerDelta(a.state, b.state) else: return S.Zero if isinstance(a, FermionicOperator) and isinstance(b, FermionicOperator): return wicks(a*b) - wicks(b*a) # # Canonical ordering of arguments # if a.sort_key() > b.sort_key(): return S.NegativeOne*cls(b, a)
[docs] def doit(self, **hints): """ Enables the computation of complex expressions. Examples ======== >>> from sympy.physics.secondquant import Commutator, F, Fd >>> from sympy import symbols >>> i, j = symbols('i,j', below_fermi=True) >>> a, b = symbols('a,b', above_fermi=True) >>> c = Commutator(Fd(a)*F(i),Fd(b)*F(j)) >>> c.doit(wicks=True) 0 """ a = self.args[0] b = self.args[1] if hints.get("wicks"): a = a.doit(**hints) b = b.doit(**hints) try: return wicks(a*b) - wicks(b*a) except ContractionAppliesOnlyToFermions: pass except WicksTheoremDoesNotApply: pass return (a*b - b*a).doit(**hints)
def __repr__(self): return "Commutator(%s,%s)" % (self.args[0], self.args[1]) def __str__(self): return "[%s,%s]" % (self.args[0], self.args[1]) def _latex(self, printer): return "\\left[%s,%s\\right]" % tuple([ printer._print(arg) for arg in self.args])
[docs]class NO(Expr): """ This Object is used to represent normal ordering brackets. i.e. {abcd} sometimes written :abcd: Applying the function NO(arg) to an argument means that all operators in the argument will be assumed to anticommute, and have vanishing contractions. This allows an immediate reordering to canonical form upon object creation. >>> from sympy import symbols >>> from sympy.physics.secondquant import NO, F, Fd >>> p,q = symbols('p,q') >>> NO(Fd(p)*F(q)) NO(CreateFermion(p)*AnnihilateFermion(q)) >>> NO(F(q)*Fd(p)) -NO(CreateFermion(p)*AnnihilateFermion(q)) Note: If you want to generate a normal ordered equivalent of an expression, you should use the function wicks(). This class only indicates that all operators inside the brackets anticommute, and have vanishing contractions. Nothing more, nothing less. """ is_commutative = False def __new__(cls, arg): """ Use anticommutation to get canonical form of operators. Employ associativity of normal ordered product: {ab{cd}} = {abcd} but note that {ab}{cd} /= {abcd}. We also employ distributivity: {ab + cd} = {ab} + {cd}. Canonical form also implies expand() {ab(c+d)} = {abc} + {abd}. """ # {ab + cd} = {ab} + {cd} arg = sympify(arg) arg = arg.expand() if arg.is_Add: return Add(*[ cls(term) for term in arg.args]) if arg.is_Mul: # take coefficient outside of normal ordering brackets c_part, seq = arg.args_cnc() if c_part: coeff = Mul(*c_part) if not seq: return coeff else: coeff = S.One # {ab{cd}} = {abcd} newseq = [] foundit = False for fac in seq: if isinstance(fac, NO): newseq.extend(fac.args) foundit = True else: newseq.append(fac) if foundit: return coeff*cls(Mul(*newseq)) # We assume that the user don't mix B and F operators if isinstance(seq[0], BosonicOperator): raise NotImplementedError try: newseq, sign = _sort_anticommuting_fermions(seq) except ViolationOfPauliPrinciple: return S.Zero if sign % 2: return (S.NegativeOne*coeff)*cls(Mul(*newseq)) elif sign: return coeff*cls(Mul(*newseq)) else: pass # since sign==0, no permutations was necessary # if we couldn't do anything with Mul object, we just # mark it as normal ordered if coeff != S.One: return coeff*cls(Mul(*newseq)) return Expr.__new__(cls, Mul(*newseq)) if isinstance(arg, NO): return arg # if object was not Mul or Add, normal ordering does not apply return arg @property
[docs] def has_q_creators(self): """ Return 0 if the leftmost argument of the first argument is a not a q_creator, else 1 if it is above fermi or -1 if it is below fermi. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import NO, F, Fd >>> a = symbols('a', above_fermi=True) >>> i = symbols('i', below_fermi=True) >>> NO(Fd(a)*Fd(i)).has_q_creators 1 >>> NO(F(i)*F(a)).has_q_creators -1 >>> NO(Fd(i)*F(a)).has_q_creators #doctest: +SKIP 0 """ return self.args[0].args[0].is_q_creator
@property
[docs] def has_q_annihilators(self): """ Return 0 if the rightmost argument of the first argument is a not a q_annihilator, else 1 if it is above fermi or -1 if it is below fermi. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import NO, F, Fd >>> a = symbols('a', above_fermi=True) >>> i = symbols('i', below_fermi=True) >>> NO(Fd(a)*Fd(i)).has_q_annihilators -1 >>> NO(F(i)*F(a)).has_q_annihilators 1 >>> NO(Fd(a)*F(i)).has_q_annihilators 0 """ return self.args[0].args[-1].is_q_annihilator
[docs] def doit(self, **kw_args): """ Either removes the brackets or enables complex computations in its arguments. Examples ======== >>> from sympy.physics.secondquant import NO, Fd, F >>> from textwrap import fill >>> from sympy import symbols, Dummy >>> p,q = symbols('p,q', cls=Dummy) >>> print(fill(str(NO(Fd(p)*F(q)).doit()))) KroneckerDelta(_a, _p)*KroneckerDelta(_a, _q)*CreateFermion(_a)*AnnihilateFermion(_a) + KroneckerDelta(_a, _p)*KroneckerDelta(_i, _q)*CreateFermion(_a)*AnnihilateFermion(_i) - KroneckerDelta(_a, _q)*KroneckerDelta(_i, _p)*AnnihilateFermion(_a)*CreateFermion(_i) - KroneckerDelta(_i, _p)*KroneckerDelta(_i, _q)*AnnihilateFermion(_i)*CreateFermion(_i) """ if kw_args.get("remove_brackets", True): return self._remove_brackets() else: return self.__new__(type(self), self.args[0].doit(**kw_args))
def _remove_brackets(self): """ Returns the sorted string without normal order brackets. The returned string have the property that no nonzero contractions exist. """ # check if any creator is also an annihilator subslist = [] for i in self.iter_q_creators(): if self[i].is_q_annihilator: assume = self[i].state.assumptions0 # only operators with a dummy index can be split in two terms if isinstance(self[i].state, Dummy): # create indices with fermi restriction assume.pop("above_fermi", None) assume["below_fermi"] = True below = Dummy('i', **assume) assume.pop("below_fermi", None) assume["above_fermi"] = True above = Dummy('a', **assume) cls = type(self[i]) split = ( self[i].__new__(cls, below) * KroneckerDelta(below, self[i].state) + self[i].__new__(cls, above) * KroneckerDelta(above, self[i].state) ) subslist.append((self[i], split)) else: raise SubstitutionOfAmbigousOperatorFailed(self[i]) if subslist: result = NO(self.subs(subslist)) if isinstance(result, Add): return Add(*[term.doit() for term in result.args]) else: return self.args[0] def _expand_operators(self): """ Returns a sum of NO objects that contain no ambiguous q-operators. If an index q has range both above and below fermi, the operator F(q) is ambiguous in the sense that it can be both a q-creator and a q-annihilator. If q is dummy, it is assumed to be a summation variable and this method rewrites it into a sum of NO terms with unambiguous operators: {Fd(p)*F(q)} = {Fd(a)*F(b)} + {Fd(a)*F(i)} + {Fd(j)*F(b)} -{F(i)*Fd(j)} where a,b are above and i,j are below fermi level. """ return NO(self._remove_brackets) def __getitem__(self, i): if isinstance(i, slice): indices = i.indices(len(self)) return [self.args[0].args[i] for i in range(*indices)] else: return self.args[0].args[i] def __len__(self): return len(self.args[0].args)
[docs] def iter_q_annihilators(self): """ Iterates over the annihilation operators. Examples ======== >>> from sympy import symbols >>> i, j = symbols('i j', below_fermi=True) >>> a, b = symbols('a b', above_fermi=True) >>> from sympy.physics.secondquant import NO, F, Fd >>> no = NO(Fd(a)*F(i)*F(b)*Fd(j)) >>> no.iter_q_creators() <generator object... at 0x...> >>> list(no.iter_q_creators()) [0, 1] >>> list(no.iter_q_annihilators()) [3, 2] """ ops = self.args[0].args iter = xrange(len(ops) - 1, -1, -1) for i in iter: if ops[i].is_q_annihilator: yield i else: break
[docs] def iter_q_creators(self): """ Iterates over the creation operators. Examples ======== >>> from sympy import symbols >>> i, j = symbols('i j', below_fermi=True) >>> a, b = symbols('a b', above_fermi=True) >>> from sympy.physics.secondquant import NO, F, Fd >>> no = NO(Fd(a)*F(i)*F(b)*Fd(j)) >>> no.iter_q_creators() <generator object... at 0x...> >>> list(no.iter_q_creators()) [0, 1] >>> list(no.iter_q_annihilators()) [3, 2] """ ops = self.args[0].args iter = xrange(0, len(ops)) for i in iter: if ops[i].is_q_creator: yield i else: break
[docs] def get_subNO(self, i): """ Returns a NO() without FermionicOperator at index i. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import F, NO >>> p,q,r = symbols('p,q,r') >>> NO(F(p)*F(q)*F(r)).get_subNO(1) # doctest: +SKIP NO(AnnihilateFermion(p)*AnnihilateFermion(r)) """ arg0 = self.args[0] # it's a Mul by definition of how it's created mul = arg0._new_rawargs(arg0.args[:i] + arg0.args[i + 1:]) return NO(mul)
def _latex(self, printer): return "\\left\\{%s\\right\\}" % printer._print(self.args[0]) def __repr__(self): return "NO(%s)" % self.args[0] def __str__(self): return ":%s:" % self.args[0] # @cacheit
[docs]def contraction(a, b): """ Calculates contraction of Fermionic operators a and b. Examples ======== >>> from sympy import symbols >>> from sympy.physics.secondquant import F, Fd, contraction >>> p, q = symbols('p,q') >>> a, b = symbols('a,b', above_fermi=True) >>> i, j = symbols('i,j', below_fermi=True) A contraction is non-zero only if a quasi-creator is to the right of a quasi-annihilator: >>> contraction(F(a),Fd(b)) KroneckerDelta(a, b) >>> contraction(Fd(i),F(j)) KroneckerDelta(i, j) For general indices a non-zero result restricts the indices to below/above the fermi surface: >>> contraction(Fd(p),F(q)) KroneckerDelta(_i, q)*KroneckerDelta(p, q) >>> contraction(F(p),Fd(q)) KroneckerDelta(_a, q)*KroneckerDelta(p, q) Two creators or two annihilators always vanishes: >>> contraction(F(p),F(q)) 0 >>> contraction(Fd(p),Fd(q)) 0 """ if isinstance(b, FermionicOperator) and isinstance(a, FermionicOperator): if isinstance(a, AnnihilateFermion) and isinstance(b, CreateFermion): if b.state.assumptions0.get("below_fermi"): return S.Zero if a.state.assumptions0.get("below_fermi"): return S.Zero if b.state.assumptions0.get("above_fermi"): return KroneckerDelta(a.state, b.state) if a.state.assumptions0.get("above_fermi"): return KroneckerDelta(a.state, b.state) return (KroneckerDelta(a.state, b.state)* KroneckerDelta(b.state, Dummy('a', above_fermi=True))) if isinstance(b, AnnihilateFermion) and isinstance(a, CreateFermion): if b.state.assumptions0.get("above_fermi"): return S.Zero if a.state.assumptions0.get("above_fermi"): return S.Zero if b.state.assumptions0.get("below_fermi"): return KroneckerDelta(a.state, b.state) if a.state.assumptions0.get("below_fermi"): return KroneckerDelta(a.state, b.state) return (KroneckerDelta(a.state, b.state)* KroneckerDelta(b.state, Dummy('i', below_fermi=True))) # vanish if 2xAnnihilator or 2xCreator return S.Zero else: #not fermion operators t = ( isinstance(i, FermionicOperator) for i in (a, b) ) raise ContractionAppliesOnlyToFermions(*t)
def _sqkey(sq_operator): """Generates key for canonical sorting of SQ operators.""" return sq_operator._sortkey() def _sort_anticommuting_fermions(string1, key=_sqkey): """Sort fermionic operators to canonical order, assuming all pairs anticommute. Uses a bidirectional bubble sort. Items in string1 are not referenced so in principle they may be any comparable objects. The sorting depends on the operators '>' and '=='. If the Pauli principle is violated, an exception is raised. Returns ======= tuple (sorted_str, sign) sorted_str: list containing the sorted operators sign: int telling how many times the sign should be changed (if sign==0 the string was already sorted) """ verified = False sign = 0 rng = list(range(len(string1) - 1)) rev = list(range(len(string1) - 3, -1, -1)) keys = list(map(key, string1)) key_val = dict(list(zip(keys, string1))) while not verified: verified = True for i in rng: left = keys[i] right = keys[i + 1] if left == right: raise ViolationOfPauliPrinciple([left, right]) if left > right: verified = False keys[i:i + 2] = [right, left] sign = sign + 1 if verified: break for i in rev: left = keys[i] right = keys[i + 1] if left == right: raise ViolationOfPauliPrinciple([left, right]) if left > right: verified = False keys[i:i + 2] = [right, left] sign = sign + 1 string1 = [ key_val[k] for k in keys ] return (string1, sign)
[docs]def evaluate_deltas(e): """ We evaluate KroneckerDelta symbols in the expression assuming Einstein summation. If one index is repeated it is summed over and in effect substituted with the other one. If both indices are repeated we substitute according to what is the preferred index. this is determined by KroneckerDelta.preferred_index and KroneckerDelta.killable_index. In case there are no possible substitutions or if a substitution would imply a loss of information, nothing is done. In case an index appears in more than one KroneckerDelta, the resulting substitution depends on the order of the factors. Since the ordering is platform dependent, the literal expression resulting from this function may be hard to predict. Examples ======== We assume the following: >>> from sympy import symbols, Function, Dummy, KroneckerDelta >>> from sympy.physics.secondquant import evaluate_deltas >>> i,j = symbols('i j', below_fermi=True, cls=Dummy) >>> a,b = symbols('a b', above_fermi=True, cls=Dummy) >>> p,q = symbols('p q', cls=Dummy) >>> f = Function('f') >>> t = Function('t') The order of preference for these indices according to KroneckerDelta is (a, b, i, j, p, q). Trivial cases: >>> evaluate_deltas(KroneckerDelta(i,j)*f(i)) # d_ij f(i) -> f(j) f(_j) >>> evaluate_deltas(KroneckerDelta(i,j)*f(j)) # d_ij f(j) -> f(i) f(_i) >>> evaluate_deltas(KroneckerDelta(i,p)*f(p)) # d_ip f(p) -> f(i) f(_i) >>> evaluate_deltas(KroneckerDelta(q,p)*f(p)) # d_qp f(p) -> f(q) f(_q) >>> evaluate_deltas(KroneckerDelta(q,p)*f(q)) # d_qp f(q) -> f(p) f(_p) More interesting cases: >>> evaluate_deltas(KroneckerDelta(i,p)*t(a,i)*f(p,q)) f(_i, _q)*t(_a, _i) >>> evaluate_deltas(KroneckerDelta(a,p)*t(a,i)*f(p,q)) f(_a, _q)*t(_a, _i) >>> evaluate_deltas(KroneckerDelta(p,q)*f(p,q)) f(_p, _p) Finally, here are some cases where nothing is done, because that would imply a loss of information: >>> evaluate_deltas(KroneckerDelta(i,p)*f(q)) f(_q)*KroneckerDelta(_i, _p) >>> evaluate_deltas(KroneckerDelta(i,p)*f(i)) f(_i)*KroneckerDelta(_i, _p) """ # We treat Deltas only in mul objects # for general function objects we don't evaluate KroneckerDeltas in arguments, # but here we hard code exceptions to this rule accepted_functions = ( Add, ) if isinstance(e, accepted_functions): return e.func(*[evaluate_deltas(arg) for arg in e.args]) elif isinstance(e, Mul): # find all occurences of delta function and count each index present in # expression. deltas = [] indices = {} for i in e.args: for s in i.atoms(): if s in indices: indices[s] += 1 else: indices[s] = 0 # geek counting simplifies logic below if isinstance(i, KroneckerDelta): deltas.append(i) for d in deltas: # If we do something, and there are more deltas, we should recurse # to treat the resulting expression properly if indices[d.killable_index]: e = e.subs(d.killable_index, d.preferred_index) if len(deltas) > 1: return evaluate_deltas(e) elif indices[d.preferred_index] and d.indices_contain_equal_information: e = e.subs(d.preferred_index, d.killable_index) if len(deltas) > 1: return evaluate_deltas(e) else: pass return e # nothing to do, maybe we hit a Symbol or a number else: return e
[docs]def substitute_dummies(expr, new_indices=False, pretty_indices={}): """ Collect terms by substitution of dummy variables. This routine allows simplification of Add expressions containing terms which differ only due to dummy variables. The idea is to substitute all dummy variables consistently depending on the structure of the term. For each term, we obtain a sequence of all dummy variables, where the order is determined by the index range, what factors the index belongs to and its position in each factor. See _get_ordered_dummies() for more inforation about the sorting of dummies. The index sequence is then substituted consistently in each term. Examples ======== >>> from sympy import symbols, Function, Dummy >>> from sympy.physics.secondquant import substitute_dummies >>> a,b,c,d = symbols('a b c d', above_fermi=True, cls=Dummy) >>> i,j = symbols('i j', below_fermi=True, cls=Dummy) >>> f = Function('f') >>> expr = f(a,b) + f(c,d); expr f(_a, _b) + f(_c, _d) Since a, b, c and d are equivalent summation indices, the expression can be simplified to a single term (for which the dummy indices are still summed over) >>> substitute_dummies(expr) 2*f(_a, _b) Controlling output: By default the dummy symbols that are already present in the expression will be reused in a different permuation. However, if new_indices=True, new dummies will be generated and inserted. The keyword 'pretty_indices' can be used to control this generation of new symbols. By default the new dummies will be generated on the form i_1, i_2, a_1, etc. If you supply a dictionary with key:value pairs in the form: { index_group: string_of_letters } The letters will be used as labels for the new dummy symbols. The index_groups must be one of 'above', 'below' or 'general'. >>> expr = f(a,b,i,j) >>> my_dummies = { 'above':'st', 'below':'uv' } >>> substitute_dummies(expr, new_indices=True, pretty_indices=my_dummies) f(_s, _t, _u, _v) If we run out of letters, or if there is no keyword for some index_group the default dummy generator will be used as a fallback: >>> p,q = symbols('p q', cls=Dummy) # general indices >>> expr = f(p,q) >>> substitute_dummies(expr, new_indices=True, pretty_indices=my_dummies) f(_p_0, _p_1) """ # setup the replacing dummies if new_indices: letters_above = pretty_indices.get('above', "") letters_below = pretty_indices.get('below', "") letters_general = pretty_indices.get('general', "") len_above = len(letters_above) len_below = len(letters_below) len_general = len(letters_general) def _i(number): try: return letters_below[number] except IndexError: return 'i_' + str(number - len_below) def _a(number): try: return letters_above[number] except IndexError: return 'a_' + str(number - len_above) def _p(number): try: return letters_general[number] except IndexError: return 'p_' + str(number - len_general) aboves = [] belows = [] generals = [] dummies = expr.atoms(Dummy) if not new_indices: dummies = sorted(dummies, key=default_sort_key) # generate lists with the dummies we will insert a = i = p = 0 for d in dummies: assum = d.assumptions0 if assum.get("above_fermi"): if new_indices: sym = _a(a) a += 1 l1 = aboves elif assum.get("below_fermi"): if new_indices: sym = _i(i) i += 1 l1 = belows else: if new_indices: sym = _p(p) p += 1 l1 = generals if new_indices: l1.append(Dummy(sym, **assum)) else: l1.append(d) expr = expr.expand() terms = Add.make_args(expr) new_terms = [] for term in terms: i = iter(belows) a = iter(aboves) p = iter(generals) ordered = _get_ordered_dummies(term) subsdict = {} for d in ordered: if d.assumptions0.get('below_fermi'): subsdict[d] = next(i) elif d.assumptions0.get('above_fermi'): subsdict[d] = next(a) else: subsdict[d] = next(p) subslist = [] final_subs = [] for k, v in subsdict.items(): if k == v: continue if v in subsdict: # We check if the sequence of substitutions end quickly. In # that case, we can avoid temporary symbols if we ensure the # correct substitution order. if subsdict[v] in subsdict: # (x, y) -> (y, x), we need a temporary variable x = Dummy('x') subslist.append((k, x)) final_subs.append((x, v)) else: # (x, y) -> (y, a), x->y must be done last # but before temporary variables are resolved final_subs.insert(0, (k, v)) else: subslist.append((k, v)) subslist.extend(final_subs) new_terms.append(term.subs(subslist)) return Add(*new_terms)
class KeyPrinter(StrPrinter): """Printer for which only equal objects are equal in print""" def _print_Dummy(self, expr): return "(%s_%i)" % (expr.name, expr.dummy_index) def __kprint(expr): p = KeyPrinter() return p.doprint(expr) def _get_ordered_dummies(mul, verbose=False): """Returns all dummies in the mul sorted in canonical order The purpose of the canonical ordering is that dummies can be substituted consistently across terms with the result that equivalent terms can be simplified. It is not possible to determine if two terms are equivalent based solely on the dummy order. However, a consistent substitution guided by the ordered dummies should lead to trivially (non-)equivalent terms, thereby revealing the equivalence. This also means that if two terms have identical sequences of dummies, the (non-)equivalence should already be apparent. Strategy -------- The canoncial order is given by an arbitrary sorting rule. A sort key is determined for each dummy as a tuple that depends on all factors where the index is present. The dummies are thereby sorted according to the contraction structure of the term, instead of sorting based solely on the dummy symbol itself. After all dummies in the term has been assigned a key, we check for identical keys, i.e. unorderable dummies. If any are found, we call a specialized method, _determine_ambiguous(), that will determine a unique order based on recursive calls to _get_ordered_dummies(). Key description --------------- A high level description of the sort key: 1. Range of the dummy index 2. Relation to external (non-dummy) indices 3. Position of the index in the first factor 4. Position of the index in the second factor The sort key is a tuple with the following components: 1. A single character indicating the range of the dummy (above, below or general.) 2. A list of strings with fully masked string representations of all factors where the dummy is present. By masked, we mean that dummies are represented by a symbol to indicate either below fermi, above or general. No other information is displayed about the dummies at this point. The list is sorted stringwise. 3. An integer number indicating the position of the index, in the first factor as sorted in 2. 4. An integer number indicating the position of the index, in the second factor as sorted in 2. If a factor is either of type AntiSymmetricTensor or SqOperator, the index position in items 3 and 4 is indicated as 'upper' or 'lower' only. (Creation operators are considered upper and annihilation operators lower.) If the masked factors are identical, the two factors cannot be ordered unambiguously in item 2. In this case, items 3, 4 are left out. If several indices are contracted between the unorderable factors, it will be handled by _determine_ambiguous() """ # setup dicts to avoid repeated calculations in key() args = Mul.make_args(mul) fac_dum = dict([ (fac, fac.atoms(Dummy)) for fac in args] ) fac_repr = dict([ (fac, __kprint(fac)) for fac in args] ) all_dums = reduce(set.union, list(fac_dum.values()), set()) mask = {} for d in all_dums: if d.assumptions0.get('below_fermi'): mask[d] = '0' elif d.assumptions0.get('above_fermi'): mask[d] = '1' else: mask[d] = '2' dum_repr = dict([ (d, __kprint(d)) for d in all_dums ]) def _key(d): dumstruct = [ fac for fac in fac_dum if d in fac_dum[fac] ] other_dums = reduce( set.union, [ fac_dum[fac] for fac in dumstruct ], set()) fac = dumstruct[-1] if other_dums is fac_dum[fac]: other_dums = fac_dum[fac].copy() other_dums.remove(d) masked_facs = [ fac_repr[fac] for fac in dumstruct ] for d2 in other_dums: masked_facs = [ fac.replace(dum_repr[d2], mask[d2]) for fac in masked_facs ] all_masked = [ fac.replace(dum_repr[d], mask[d]) for fac in masked_facs ] masked_facs = dict(list(zip(dumstruct, masked_facs))) # dummies for which the ordering cannot be determined if has_dups(all_masked): all_masked.sort() return mask[d], tuple(all_masked) # positions are ambiguous # sort factors according to fully masked strings keydict = dict(list(zip(dumstruct, all_masked))) dumstruct.sort(key=lambda x: keydict[x]) all_masked.sort() pos_val = [] for fac in dumstruct: if isinstance(fac, AntiSymmetricTensor): if d in fac.upper: pos_val.append('u') if d in fac.lower: pos_val.append('l') elif isinstance(fac, Creator): pos_val.append('u') elif isinstance(fac, Annihilator): pos_val.append('l') elif isinstance(fac, NO): ops = [ op for op in fac if op.has(d) ] for op in ops: if isinstance(op, Creator): pos_val.append('u') else: pos_val.append('l') else: # fallback to position in string representation facpos = -1 while 1: facpos = masked_facs[fac].find(dum_repr[d], facpos + 1) if facpos == -1: break pos_val.append(facpos) return (mask[d], tuple(all_masked), pos_val[0], pos_val[-1]) dumkey = dict(list(zip(all_dums, list(map(_key, all_dums))))) result = sorted(all_dums, key=lambda x: dumkey[x]) if has_dups(iter(dumkey.values())): # We have ambiguities unordered = defaultdict(set) for d, k in dumkey.items(): unordered[k].add(d) for k in [ k for k in unordered if len(unordered[k]) < 2 ]: del unordered[k] unordered = [ unordered[k] for k in sorted(unordered) ] result = _determine_ambiguous(mul, result, unordered) return result def _determine_ambiguous(term, ordered, ambiguous_groups): # We encountered a term for which the dummy substitution is ambiguous. # This happens for terms with 2 or more contractions between factors that # cannot be uniquely ordered independent of summation indices. For # example: # # Sum(p, q) v^{p, .}_{q, .}v^{q, .}_{p, .} # # Assuming that the indices represented by . are dummies with the # same range, the factors cannot be ordered, and there is no # way to determine a consistent ordering of p and q. # # The strategy employed here, is to relabel all unambiguous dummies with # non-dummy symbols and call _get_ordered_dummies again. This procedure is # applied to the entire term so there is a possibility that # _determine_ambiguous() is called again from a deeper recursion level. # break recursion if there are no ordered dummies all_ambiguous = set() for dummies in ambiguous_groups: all_ambiguous |= dummies all_ordered = set(ordered) - all_ambiguous if not all_ordered: # FIXME: If we arrive here, there are no ordered dummies. A method to # handle this needs to be implemented. In order to return something # useful nevertheless, we choose arbitrarily the first dummy and # determine the rest from this one. This method is dependent on the # actual dummy labels which violates an assumption for the canonization # procedure. A better implementation is needed. group = [ d for d in ordered if d in ambiguous_groups[0] ] d = group[0] all_ordered.add(d) ambiguous_groups[0].remove(d) stored_counter = _symbol_factory._counter subslist = [] for d in [ d for d in ordered if d in all_ordered ]: nondum = _symbol_factory._next() subslist.append((d, nondum)) newterm = term.subs(subslist) neworder = _get_ordered_dummies(newterm) _symbol_factory._set_counter(stored_counter) # update ordered list with new information for group in ambiguous_groups: ordered_group = [ d for d in neworder if d in group ] ordered_group.reverse() result = [] for d in ordered: if d in group: result.append(ordered_group.pop()) else: result.append(d) ordered = result return ordered class _SymbolFactory(object): def __init__(self, label): self._counterVar = 0 self._label = label def _set_counter(self, value): """ Sets counter to value. """ self._counterVar = value @property def _counter(self): """ What counter is currently at. """ return self._counterVar def _next(self): """ Generates the next symbols and increments counter by 1. """ s = Symbol("%s%i" % (self._label, self._counterVar)) self._counterVar += 1 return s _symbol_factory = _SymbolFactory('_]"]_') # most certainly a unique label @cacheit def _get_contractions(string1, keep_only_fully_contracted=False): """ Returns Add-object with contracted terms. Uses recursion to find all contractions. -- Internal helper function -- Will find nonzero contractions in string1 between indices given in leftrange and rightrange. """ # Should we store current level of contraction? if keep_only_fully_contracted and string1: result = [] else: result = [NO(Mul(*string1))] for i in range(len(string1) - 1): for j in range(i + 1, len(string1)): c = contraction(string1[i], string1[j]) if c: # print "found contraction",c sign = (j - i + 1) % 2 if sign: coeff = S.NegativeOne*c else: coeff = c # # Call next level of recursion # ============================ # # We now need to find more contractions among operators # # oplist = string1[:i]+ string1[i+1:j] + string1[j+1:] # # To prevent overcounting, we don't allow contractions # we have already encountered. i.e. contractions between # string1[:i] <---> string1[i+1:j] # and string1[:i] <---> string1[j+1:]. # # This leaves the case: oplist = string1[i + 1:j] + string1[j + 1:] if oplist: result.append(coeff*NO( Mul(*string1[:i])*_get_contractions( oplist, keep_only_fully_contracted=keep_only_fully_contracted))) else: result.append(coeff*NO( Mul(*string1[:i]))) if keep_only_fully_contracted: break # next iteration over i leaves leftmost operator string1[0] uncontracted return Add(*result) # @cacheit
[docs]def wicks(e, **kw_args): """ Returns the normal ordered equivalent of an expression using Wicks Theorem. Examples ======== >>> from sympy import symbols, Function, Dummy >>> from sympy.physics.secondquant import wicks, F, Fd, NO >>> p,q,r = symbols('p,q,r') >>> wicks(Fd(p)*F(q)) # doctest: +SKIP d(p, q)*d(q, _i) + NO(CreateFermion(p)*AnnihilateFermion(q)) By default, the expression is expanded: >>> wicks(F(p)*(F(q)+F(r))) # doctest: +SKIP NO(AnnihilateFermion(p)*AnnihilateFermion(q)) + NO( AnnihilateFermion(p)*AnnihilateFermion(r)) With the keyword 'keep_only_fully_contracted=True', only fully contracted terms are returned. By request, the result can be simplified in the following order: -- KroneckerDelta functions are evaluated -- Dummy variables are substituted consistently across terms >>> p, q, r = symbols('p q r', cls=Dummy) >>> wicks(Fd(p)*(F(q)+F(r)), keep_only_fully_contracted=True) # doctest: +SKIP KroneckerDelta(_i, _q)*KroneckerDelta( _p, _q) + KroneckerDelta(_i, _r)*KroneckerDelta(_p, _r) """ if not e: return S.Zero opts = { 'simplify_kronecker_deltas': False, 'expand': True, 'simplify_dummies': False, 'keep_only_fully_contracted': False } opts.update(kw_args) # check if we are already normally ordered if isinstance(e, NO): if opts['keep_only_fully_contracted']: return S.Zero else: return e elif isinstance(e, FermionicOperator): if opts['keep_only_fully_contracted']: return S.Zero else: return e # break up any NO-objects, and evaluate commutators e = e.doit(wicks=True) # make sure we have only one term to consider e = e.expand() if isinstance(e, Add): if opts['simplify_dummies']: return substitute_dummies(Add(*[ wicks(term, **kw_args) for term in e.args])) else: return Add(*[ wicks(term, **kw_args) for term in e.args]) # For Mul-objects we can actually do something if isinstance(e, Mul): # we dont want to mess around with commuting part of Mul # so we factorize it out before starting recursion c_part = [] string1 = [] for factor in e.args: if factor.is_commutative: c_part.append(factor) else: string1.append(factor) n = len(string1) # catch trivial cases if n == 0: result = e elif n == 1: if opts['keep_only_fully_contracted']: return S.Zero else: result = e else: # non-trivial if isinstance(string1[0], BosonicOperator): raise NotImplementedError string1 = tuple(string1) # recursion over higher order contractions result = _get_contractions(string1, keep_only_fully_contracted=opts['keep_only_fully_contracted'] ) result = Mul(*c_part)*result if opts['expand']: result = result.expand() if opts['simplify_kronecker_deltas']: result = evaluate_deltas(result) return result # there was nothing to do return e
[docs]class PermutationOperator(Expr): """ Represents the index permutation operator P(ij). P(ij)*f(i)*g(j) = f(i)*g(j) - f(j)*g(i) """ is_commutative = True def __new__(cls, i, j): i, j = sorted(map(sympify, (i, j)), key=default_sort_key) obj = Basic.__new__(cls, i, j) return obj
[docs] def get_permuted(self, expr): """ Returns -expr with permuted indices. >>> from sympy import symbols, Function >>> from sympy.physics.secondquant import PermutationOperator >>> p,q = symbols('p,q') >>> f = Function('f') >>> PermutationOperator(p,q).get_permuted(f(p,q)) -f(q, p) """ i = self.args[0] j = self.args[1] if expr.has(i) and expr.has(j): tmp = Dummy() expr = expr.subs(i, tmp) expr = expr.subs(j, i) expr = expr.subs(tmp, j) return S.NegativeOne*expr else: return expr
def _latex(self, printer): return "P(%s%s)" % self.args
[docs]def simplify_index_permutations(expr, permutation_operators): """ Performs simplification by introducing PermutationOperators where appropriate. Schematically: [abij] - [abji] - [baij] + [baji] -> P(ab)*P(ij)*[abij] permutation_operators is a list of PermutationOperators to consider. If permutation_operators=[P(ab),P(ij)] we will try to introduce the permutation operators P(ij) and P(ab) in the expression. If there are other possible simplifications, we ignore them. >>> from sympy import symbols, Function >>> from sympy.physics.secondquant import simplify_index_permutations >>> from sympy.physics.secondquant import PermutationOperator >>> p,q,r,s = symbols('p,q,r,s') >>> f = Function('f') >>> g = Function('g') >>> expr = f(p)*g(q) - f(q)*g(p); expr f(p)*g(q) - f(q)*g(p) >>> simplify_index_permutations(expr,[PermutationOperator(p,q)]) f(p)*g(q)*PermutationOperator(p, q) >>> PermutList = [PermutationOperator(p,q),PermutationOperator(r,s)] >>> expr = f(p,r)*g(q,s) - f(q,r)*g(p,s) + f(q,s)*g(p,r) - f(p,s)*g(q,r) >>> simplify_index_permutations(expr,PermutList) f(p, r)*g(q, s)*PermutationOperator(p, q)*PermutationOperator(r, s) """ def _get_indices(expr, ind): """ Collects indices recursively in predictable order. """ result = [] for arg in expr.args: if arg in ind: result.append(arg) else: if arg.args: result.extend(_get_indices(arg, ind)) return result def _choose_one_to_keep(a, b, ind): # we keep the one where indices in ind are in order ind[0] < ind[1] return min(a, b, key=lambda x: default_sort_key(_get_indices(x, ind))) expr = expr.expand() if isinstance(expr, Add): terms = set(expr.args) for P in permutation_operators: new_terms = set([]) on_hold = set([]) while terms: term = terms.pop() permuted = P.get_permuted(term) if permuted in terms | on_hold: try: terms.remove(permuted) except KeyError: on_hold.remove(permuted) keep = _choose_one_to_keep(term, permuted, P.args) new_terms.add(P*keep) else: # Some terms must get a second chance because the permuted # term may already have canonical dummy ordering. Then # substitute_dummies() does nothing. However, the other # term, if it exists, will be able to match with us. permuted1 = permuted permuted = substitute_dummies(permuted) if permuted1 == permuted: on_hold.add(term) elif permuted in terms | on_hold: try: terms.remove(permuted) except KeyError: on_hold.remove(permuted) keep = _choose_one_to_keep(term, permuted, P.args) new_terms.add(P*keep) else: new_terms.add(term) terms = new_terms | on_hold return Add(*terms) return expr