This Physics Library got introduced in Maple 17 with 96 programming commands and 58 specialized Physics types, all key elements of the programming language used at Maplesoft to construct Physics. This library has been growing; in Maple 2019 there are 159 programming commands and 98 Physics types. These commands and types can be useful to program your own extensions of Physics. Many of these commands are useful interactively too, like for instance AntiCommute, Commute, Degree for noncommutative variables, IsTensorialAntisymmetric, IsTensorialSymmetric, MinimizeTensorComponents for tensors, Backup and Restore to explore with different settings for algebra rules, spacetime dimensions, and so on.

To load this Library of programming commands, enter:

with(Physics): with(Library):

You can also use any of these commands without loading the Library by using the long form, as in Library:-Command.

To see a complete list of the Physics types available, because PhysicsType is one of the commands of the Library, and a module itself, you can enter exports(PhysicsType), or instead list them alphabetically using another Library command, ListPhysicsTypes:

ListPhysicsTypes();

$\left[\mathrm{AnnihilationCreationOperator}\,\mathrm{AnnihilationOperator}\,\mathrm{AppliableProcedure}\,\mathrm{AppliedQuantumOperator}\,\mathrm{AtProcedure}\,\mathrm{BosonAnnihilationCreationOperator}\,\mathrm{BosonAnnihilationOperator}\,\mathrm{BosonCreationOperator}\,\mathrm{Bra}\,\mathrm{Bracket}\,\mathrm{CommutativeMapping}\,\mathrm{CompositeDifferentiationOperator}\,\mathrm{Contravariant}\,\mathrm{ContravariantGreek}\,\mathrm{Covariant}\,\mathrm{CreationOperator}\,\mathrm{D\_}\,\mathrm{DifferentialOperator}\,\mathrm{DifferentialOperatorIndexed}\,\mathrm{DifferentialOperatorSymbol}\,\mathrm{DiracSpinor}\,\mathrm{DiracSpinorWithoutIndices}\,\mathrm{DoublyIndexedPhysicsMatrix}\,\mathrm{EntirelyNotNestedIntegral}\,\mathrm{EuclideanIndex}\,\mathrm{EuclideanTensor}\,\mathrm{ExpandableProduct}\,\mathrm{ExtendedBra}\,\mathrm{ExtendedDifferentialOperator}\,\mathrm{ExtendedDifferentialOperatorIndexed}\,\mathrm{ExtendedDifferentialOperatorSymbol}\,\mathrm{ExtendedKet}\,\mathrm{ExtendedMatrix}\,\mathrm{ExtendedMatrixRequiringEval}\,\mathrm{ExtendedPhysicsMatrix}\,\mathrm{ExtendedProjector}\,\mathrm{ExtendedProjectorKind}\,\mathrm{ExtendedQuantumOperator}\,\mathrm{ExtendedTensor}\,\mathrm{ExtendedVectorDifferentialOperator}\,\mathrm{FermionAnnihilationCreationOperator}\,\mathrm{FermionAnnihilationOperator}\,\mathrm{FermionCreationOperator}\,\mathrm{FullyCommutative}\,\mathrm{HalfInteger}\,\mathrm{HermitianOperator}\,\mathrm{IdentityMatrix}\,\mathrm{InertSymbol}\,\mathrm{Ket}\,\mathrm{LeviCivitaNonGalilean}\,\mathrm{LeviCivitaTensor}\,\mathrm{LinearOperatorFunction}\,\mathrm{LiteralSubscript}\,\mathrm{NegativePower}\,\mathrm{NonCommutativeProduct}\,\mathrm{NonTensorFunction}\,\mathrm{NotNestedIntegral}\,\mathrm{NumericalIndex}\,\mathrm{P\_indexed}\,\mathrm{P\_suffixed}\,\mathrm{PhysicsD3\_2}\,\mathrm{PhysicsD\_2}\,\mathrm{PhysicsFunction}\,\mathrm{PhysicsKnownFunction}\,\mathrm{PhysicsMatrix}\,\mathrm{PhysicsMatrixWithMatrixIndices}\,\mathrm{PhysicsTensor}\,\mathrm{PhysicsVectors}\,\mathrm{Physicsd\_2}\,\mathrm{Polynomial}\,\mathrm{Procedure}\,\mathrm{Projector}\,\mathrm{ProjectorCore}\,\mathrm{ProjectorInt}\,\mathrm{ProjectorKind}\,\mathrm{ProjectorKindDot}\,\mathrm{ProjectorKindStar}\,\mathrm{ProjectorSum}\,\mathrm{QuantumNumber}\,\mathrm{QuantumObject}\,\mathrm{QuantumOperator}\,\mathrm{QuantumOperatorFunction}\,\mathrm{SpaceNumericalIndex}\,\mathrm{SpaceTimeVectorApplied}\,\mathrm{SpacetimeNumericalIndex}\,\mathrm{Spinor}\,\mathrm{SpinorWithoutIndices}\,\mathrm{SumOfNegatives}\,\mathrm{SymbolicTensorIndex}\,\mathrm{Tensor}\,\mathrm{TensorAllowWithoutIndices}\,\mathrm{TensorInCurrentCoordinates}\,\mathrm{TensorIndex}\,\mathrm{TensorInternalRepresentation}\,\mathrm{TensorQuantumOperator}\,\mathrm{TensorStructure}\,\mathrm{TensorVector}\,\mathrm{TensorWithAbstractIndices}\,\mathrm{TensorWithIndices}\,\mathrm{TensorWithNumericalIndices}\,\mathrm{TensorWithoutIndices}\,\mathrm{TensorialExpression}\,\mathrm{TripleProduct}\,\mathrm{UnitaryOperator}\,\mathrm{VectorDifferentialOperator}\,\mathrm{VectorOperator}\,\mathrm{VectorOperatorApplied}\,\mathrm{Vectorial}\,\mathrm{X\_mu}\,\mathrm{dg\_dg}\,\mathrm{genericindex}\,\mathrm{p\_P\_indexed}\,\mathrm{p\_P\_suffixed}\,\mathrm{p\_indexed}\,\mathrm{p\_suffixed}\,\mathrm{spaceindex}\,\mathrm{spacetimeindex}\,\mathrm{spinorindex}\,\mathrm{su2index}\,\mathrm{su2matrixindex}\,\mathrm{su3index}\,\mathrm{su3matrixindex}\,\mathrm{tetradindex}\,\mathrm{x\_mu}\right]$

You can see the definition of these types in two different ways. First, using the option showdefinition of ListPhysicsTypes. For example:

ListPhysicsTypes(AnnihilationCreationOperator, showdefinition);

$\left[\mathrm{AnnihilationCreationOperator}=\left(\mathrm{AnnihilationOperator}\vee \mathrm{CreationOperator}\right)\right]$

Secondly, the definition for each PhysicsType can be retrieved using TypeTools:-GetType, for example:

TypeTools:-GetType(PhysicsType:-AnnihilationCreationOperator);

$\mathrm{AnnihilationOperator}\vee \mathrm{CreationOperator}$

If you load the package of types via with(PhysicsType), you can also see this definition entering just TypeTools:-GetType(AnnihilationCreationOperator). Recall however that by entering with(PhysicsType) you will populate the space of working names with the names of all the Physics types.

These types are used as in type(A, PhysicsType:-TypeKeyword) or using the long forms Library:-PhysicsType:-TypeKeyword or Physics:-Library:-PhysicsType:-TypeKeyword, depending on whether you have already loaded Physics or Physics:-Library using with. In all cases A can be anything, and TypeKeyword is any of the types returned above by ListPhysicsTypes() or exports(PhysicsType). Note that you do not need to list these types before using them.

Mathematical notation: When Physics or Physics:-Vectors are loaded in the Standard Graphical User Interface, by entering

with(Physics): Setup(mathematicalnotation = true);

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

you have anticommutative and noncommutative variables displayed in different colors, non-projected vectors and unit vectors are respectively displayed with an arrow and a hat on top, the vectorial differential operators (such as Nabla, Laplacian) with an upside down triangle, and most of the other Physics commands are displayed as in textbooks.

Examples: illustrating the use of the package's commands in Analytical Geometry, Mechanics, Electrodynamics, Special and General Relativity, and Quantum Mechanics are found in Physics examples (this page opens only in the Standard Graphical User Interface).

Instead of providing a help page for each of these Library programming tools, below there is an alphabetical listing of them, linking to sections within this single page, where their calling sequence, parameters, description, and examples are shown.

Calling Sequence

Physics:-Library:-Add(f, k = m..n)

Parameters

f    : algebraic expression

k    : name; summation index

m, n : integers or arbitrary expressions, indicate the summation range

Description

Add unifies the standard add and sum commands into one that uses a more modern handling of arguments, is free of premature evaluation problems, and implements multi-index summation. The functionality and handling of arguments of Add can be plugged directly into the sum command using the option redefinesum of Physics:-Setup.

Examples

restart: with(Physics): with(Library):

Consider this multi-index summation.

Add(f[i, j], 1 <= i+j and i+j <= n);

$\sum _{1\le i+j\le n}{f}_{i,j}$

For instance, for $n=2$,

eval((5), n = 2);

$f_{0,1}+f_{1,0}+f_{0,2}+f_{1,1}+f_{2,0}$

The formula for the integer power of a sum is:

(a+b+c)^n = Add(factorial(n)*a^p*b^q*c^r/(factorial(p)*factorial(q)*factorial(r)), p+q+r = n);

${\left(a+b+c\right)}^n=\sum _{p+q+r=n}n!a^p{b}^q{c}^r\frac{1}{p!q!r!}$

eval((7), n = 3);

${\left(a+b+c\right)}^3=a^3+3a^{2}b+3a^{2}c+3a{b}^2+6abc+3a{c}^2+b^3+3b^{2}c+3b{c}^2+c^3$

A more modern handling of its arguments, consider a typical problem where a name, say $j$, is assigned a value, and $j$ is used as dummy summation variable

a, b, j := 1, 2, 3;

$a,b,j≔1,2,3$

The value just assigned, $j=3$, is not expected to interfere with the summation

Add(f[j], j = a..b);

$f_1+f_2$

while for this example, sum interrupts with an error message due to a premature evaluation of the summation index $j$. Another frequent case: a procedure that returns a value according to the value of the summation index.

g := k -> if k::odd then G[k] else 0 end if;

$g≔k\mapsto \mathbf{if}k::\mathrm{odd}\mathbf{then}{G}_k\mathbf{else}0\mathbf{end}\mathbf{if}$

Without giving a value to the summation limit $f$, the following summation cannot be performed.

Add(g(k), k = 1 .. f);

$\sum _{k=1}^{f}g\left(k\right)$

For this example, add interrupts with an error message while sum returns 0, again due to a premature evaluation of $g\left(k\right)$ before $k$ has a numerical value. Using a more modern handling of arguments Add returns unevaluated instead, as seen above, which permits evaluating the summation at a later moment, for instance attributing a value to $f$.

eval((12), f = 3);

$G_1+G_3$

See Also

add, sum, Physics[Setup]

Calling Sequence

Physics:-Library:-AntiCommute(a, b)

Parameters

a    : any Maple algebraic expression

b    : any Maple algebraic expression

Description

AntiCommute returns true or false according to whether a and b anticommute, that is, a*b = -b*a.

Examples

restart: with(Physics): with(Library):

Setup(anticommutativepre = A):

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

AntiCommute(A1, A2);

$\mathrm{true}$

See Also

Commute

Calling Sequence

Physics:-Library:-ApplyCommandUsingImplicitAssumptions(command, ee)

Parameters

command : a maple command to be applied to ee

ee      : a Maple object, typically an algebraic expression or equation or set of them, containing definite integrals, sums, or products

Description

ApplyCommandUsingImplicitAssumptions applies any command to expressions containing sums, integrals or products such that the command is applied to the summand (integrand or 1st argument of the product) taking into account the assumptions implicit in the summation (integration or product) range.

Any additional argument passed to ApplyCommandUsingImplicitAssumptions is passed as an optional argument for command when performing the operation.

Examples

restart: with(Physics): with(Library):

Sum((a*b)^k, k = a .. b);

$\sum _{k=a}^b{\left(ab\right)}^k$

expand will not expand the power in the summand: it does not know about $k$ being an integer (implicit assumption in the Sum structure)

expand((16));

$\sum _{k=a}^b{a}^k{b}^k$

The following makes expand be aware of the implicit assumptions on $k$ and so expand the power

ApplyCommandUsingImplicitAssumptions(expand, (16));

$\sum _{k=a}^b{a}^k{b}^k$

See Also

assuming, simplify

Calling Sequence

Physics:-Library:-ApplyProductsOfDifferentialOperators(ee)

Parameters

e : any valid Maple object, possibly an expression, that may contain products involving differential operators

Description

Given an algebraic expression ee, or any Maple object (e.g. set, list or Matrix) containing such expressions, ApplyProductsOfDifferentialOperators applies the differential operators found in the products to the operands appearing to their right.

Examples

restart: with(Physics): with(Library): with(Vectors):

Consider the commutator between a scalar $A\left(x\&comma;y\&comma;z\right)$ and the $\nabla$ operator; use CompactDisplay for convenience

CompactDisplay(A(x, y, z), B_(x, y, z));

$A\left(x\&comma;y\&comma;z\right)\mathrm{will\; now\; be\; displayed\; as}A$

$\overrightarrow{B}\left(x\&comma;y\&comma;z\right)\mathrm{will\; now\; be\; displayed\; as}\overrightarrow{B}$

Commutator(Nabla, A(x, y, z));

${\left[\nabla ,A\right]}_{-}$

The expanded form of this commutator has the correct operatorial meaning:

expand((20));

$\nabla A-A\nabla$

So all the algebra is performed taking $\nabla$ as a noncommutative operator, as we do when computing with paper and pencil. You can apply the differential operators at any time using the ApplyProductsOfDifferentialOperators

ApplyProductsOfDifferentialOperators((21));

$A_x\stackrel{\wedge }{i}+A_y\stackrel{\wedge }{j}+A_z\stackrel{\wedge }{k}-A\nabla$

The differentiation variables associated to $\nabla$ are the coordinates of the Cartesian cylindrical and spherical systems of coordinates: $\left[x\&comma;y\&comma;z\&comma;\mathrm{\rho }\&comma;\mathrm{\phi }\&comma;r\&comma;\mathrm{\theta }\right]$. The same concept is implemented for the differential operators of special and special and general relativity, d_ and D_. Define system of coordinates, $X$, then $A$ and $B$ as a tensors and for convenience use a CompactDisplay

Coordinates(X);

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\right)\right\}$

$\left\{X\right\}$

Define(A[mu], B[mu]);

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_{\mathrm{\mu }}\&comma;B_{\mathrm{\mu }}\&comma;{\mathrm{\gamma }}_{\mathrm{\mu }}\&comma;{\mathrm{\sigma }}_{\mathrm{\mu }}\&comma;{\partial }_{\mathrm{\mu }}\&comma;g_{\mathrm{\mu },\mathrm{\nu }}\&comma;{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\&comma;X_{\mathrm{\mu }}\right\}$

CompactDisplay((A, B)(X));

$A\left(\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}A$

$B\left(\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}B$

The commutator of ${\partial }_{\mathrm{\mu }}$ and $A_{\mathrm{\nu }}$ multiplied by $B_{\mathrm{\alpha }}$, representing the application of ${\partial }_{\mathrm{\mu }}$ can now be represented as

Commutator(d_[mu], A[nu](X))*B[alpha](X);

${\left[{\partial }_{\mathrm{\mu }},A_{\mathrm{\nu }}\right]}_{-}{B}_{\mathrm{\alpha }}$

expand((26));

${\partial }_{\mathrm{\mu }}{A}_{\mathrm{\nu }}{B}_{\mathrm{\alpha }}-A_{\mathrm{\nu }}{\partial }_{\mathrm{\mu }}{B}_{\mathrm{\alpha }}$

Note that to the right of ${\partial }_{\mathrm{\mu }}$ in the first term there is the product $A_{\mathrm{\nu }}{B}_{\mathrm{\alpha }}$, so that when applying the operator to the product, two terms result, one of which cancels the second term in the above:

Library:-ApplyProductsOfDifferentialOperators((27));

${\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)B_{\mathrm{\alpha }}$

ApplyProductsOfDifferentialOperators also applies differential operators that you define as such using Setup. This is particularly useful in quantum mechanics.

Setup(differentialoperators = {[p_, [x, y, z]]}, hermitianoperators = {p, x, y, z}, algebrarules = {%Commutator(x, y) = 0, %Commutator(x, z) = 0, %Commutator(y, z) = 0}, quiet);

$\left[\mathrm{algebrarules}=\left\{{\left[x,y\right]}_{-}=0\&comma;{\left[x,z\right]}_{-}=0\&comma;{\left[y,z\right]}_{-}=0\right\}\&comma;\mathrm{differentialoperators}=\left\{\left[\overrightarrow{p}\&comma;\left[x\&comma;y\&comma;z\right]\right]\right\}\&comma;\mathrm{hermitianoperators}=\left\{p\&comma;x\&comma;y\&comma;z\right\}\right]$

alias(:-X = (x, y, z)):

CompactDisplay(F(X)):

$F\left(x\&comma;y\&comma;z\right)\mathrm{will\; now\; be\; displayed\; as}F$

%Commutator(F(X), p_)*Ket(psi, X);

${\left[F,\overrightarrow{p}\right]}_{-}\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩$

Note that $\overrightarrow{p}$, $F\left(X\right)$ and the ket $\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩$ are operands in the products above and that they do not commute: we indicated that the coordinates $X$ are the differentiation variables of $\overrightarrow{p}$. This emulates what we do when computing with these operators with paper and pencil, where we represent the application of a differential operator as a product operation.

This representation can be transformed into the (traditional in computer algebra) application of the differential operator when desired, as follows:

(31) = expand((31));

${\left[F,\overrightarrow{p}\right]}_{-}\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩=F\overrightarrow{p}\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩-\overrightarrow{p}F\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩$

(31) = ApplyProductsOfDifferentialOperators(rhs((32)));

${\left[F,\overrightarrow{p}\right]}_{-}\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩=F\overrightarrow{p}\left(\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩\right)-\overrightarrow{p}\left(F\left|{\mathrm{\psi }}_{x\&comma;y\&comma;z}\right.⟩\right)$

See Also -(lead=indent) CompactDisplay, Coordinates, Define, d_, D_, Setup, Physics:-Vectors, Physics:-Vectors:-Nabla

Calling Sequence

Physics:-Library:-Backup()

Parameters

: no parameters

Description

Backup saves in memory a copy of the current Physics Setup, so that it can be restored at a later moment, during the same Maple session, using the Physics:-Library:-Restore; command. Backup returns NULL.

Examples

restart: with(Physics): with(Library):

Setup(noncommutativeprefix);

$\left[\mathrm{noncommutativeprefix}=\mathrm{none}\right]$

Backup();

Setup(noncommutativeprefix = Z);

$\left[\mathrm{noncommutativeprefix}=\left\{Z\right\}\right]$

Commute(Z1, Z2);

$\mathrm{false}$

Restore();

Setup(noncommutativeprefix);

$\left[\mathrm{noncommutativeprefix}=\mathrm{none}\right]$

Commute(Z1, Z2);

$\mathrm{true}$

See Also

Restore

Calling Sequence

Physics:-Library:-BelongToDifferentDisjointedSpaces(A, B)

Parameters

A, B      : quantum operators set as such using Setup, or Bras or Kets

Description

Given A and B as quantum operators or Bras or Kets, BelongToDifferentDisjointedSpaces returns true or false according to whether A and B below to different (disjointed) Hilbert spaces. products, correctly handling the case of noncommutative products and powers and using Glauber's formula for combining exponentials when the exponents commute with their commutator. Note that this functionality, including taking into account the possibly noncommutative character of the operands and powers in a product, is already present in the standard combine command that makes calls to CombinePowersOfSameBase, so you can directly use combine.

Examples

restart: with(Physics): with(Library):

Setup(op = {A, B});

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{op}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{quantumoperators}=\left\{A\&comma;B\right\}\right]$

A^n * A^m + exp(A) * exp(B);

$A^m{A}^n+{\&ExponentialE;}^A{\&ExponentialE;}^B$

In this expression, the exponentials in the product ${\&ExponentialE;}^A{\&ExponentialE;}^B$ cannot be combined because neither $A$ nor $B$ are commutative:

CombinePowersOfSameBase((40));

$A^{m+n}+{\&ExponentialE;}^A{\&ExponentialE;}^B$

If, however, A and B commute with their commutator,

Setup(%Commutator(A, B) = C, %Commutator(A, C) = 0, %Commutator(B, C) = 0);

$\left[\mathrm{algebrarules}=\left\{{\left[A,B\right]}_{-}=C\&comma;{\left[A,C\right]}_{-}=0\&comma;{\left[B,C\right]}_{-}=0\right\}\right]$

CombinePowersOfSameBase((40));

$A^{m+n}+{\&ExponentialE;}^{A+B+\frac{C}{2}}$

This functionality is already present in the standard combine command that makes calls to CombinePowersOfSameBase.

combine((40));

$A^{m+n}+{\&ExponentialE;}^{A+B+\frac{C}{2}}$

See Also

BelongToTheSameDisjointedSpace, Physics[Setup]

Calling Sequence

Physics:-Library:-CombinePowersOfSameBase(ee)

Parameters

ee      : a Maple object, typically an algebraic expression or equation or a set of them

Description

CombinePowersOfSameBase combines powers of same base in products, correctly handling the case of noncommutative products and powers and using Glauber's formula for combining exponentials when the exponents commute with their commutator. Note that this functionality, including taking into account the possibly noncommutative character of the operands and powers in a product, is already present in the standard combine command that makes calls to CombinePowersOfSameBase, so you can directly use combine.

Examples

restart: with(Physics): with(Library):

Setup(op = {A, B});

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{op}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{quantumoperators}=\left\{A\&comma;B\right\}\right]$

A^n * A^m + exp(A) * exp(B);

$A^m{A}^n+{\&ExponentialE;}^A{\&ExponentialE;}^B$

In this expression, the exponentials in the product ${\&ExponentialE;}^A{\&ExponentialE;}^B$ cannot be combined because neither $A$ nor $B$ are commutative:

CombinePowersOfSameBase((40));

$A^{m+n}+{\&ExponentialE;}^A{\&ExponentialE;}^B$

If, however, A and B commute with their commutator,

Setup(%Commutator(A, B) = C, %Commutator(A, C) = 0, %Commutator(B, C) = 0);

$\left[\mathrm{algebrarules}=\left\{{\left[A,B\right]}_{-}=C\&comma;{\left[A,C\right]}_{-}=0\&comma;{\left[B,C\right]}_{-}=0\right\}\right]$

CombinePowersOfSameBase((40));

$A^{m+n}+{\&ExponentialE;}^{A+B+\frac{C}{2}}$

This functionality is already present in the standard combine command that makes calls to CombinePowersOfSameBase.

combine((40));

$A^{m+n}+{\&ExponentialE;}^{A+B+\frac{C}{2}}$

See Also

combine, Physics[Commutator], Physics[Setup]

Calling Sequence

Physics:-Library:-Commute(a, b)

Parameters

a    : any Maple algebraic expression

b    : any Maple algebraic expression

Description

Commute returns true or false according to whether a and b commute, that is, $ab=ba$.

Examples

restart: with(Physics): with(Library):

Setup(noncommutativepre = A, %Commutator(A1, A2) = 0):

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{noncommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{noncommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

Commute(A1, A2);

$\mathrm{true}$

Commute(A1, A3);

$\mathrm{false}$

See Also

AntiCommute

Calling Sequence

Physics:-Library:-Contravariant(mu)

Parameters

mu    : any Maple algebraic expression

Description

Contravariant receives an algebraic expression representing a tensor index, and returns its corresponding contravariant index according to: if mu is a symbol (not already prefixed with ~), or a non-negative integer smaller or equal to the spacetime dimension, then prefix it with (that is, concatenate) ~; else if mu is a sum of such indices, then prefix all of them with ~; otherwise return mu unchanged.

Examples

restart: with(Physics): with(Library):

Contravariant(mu);

$\mathrm{~\&mu;}$

Contravariant(mu + nu);

$\mathrm{~\&mu;}+\mathrm{~\&nu;}$

Setup(dimension);

$\left[\mathrm{dimension}=4\right]$

Contravariant(3);

$\mathrm{~3}$

Contravariant(5);

$5$

See Also

Covariant, FlipCharacter, StripTilde, StripTildeFromRepeatedIndices, ToContravariant

Calling Sequence

Physics:-Library:-CoordinatesINFO()

Parameters

: no parameters

Description

CoordinatesINFO returns information about systems of spacetime coordinates set using Coordinates or Setup, as a list with four equations, where in the left-hand sides are the local variables of Physics where the information about the coordinates is saved and in the right-hand sides are the actual value of these variables. This information is of use in various parts of the library that need to handle spacetime coordinates.

Examples

restart: with(Physics): with(Library):

CoordinatesINFO();

$\left[\mathrm{Coordinates/X}=\varnothing \&comma;\mathrm{Coordinates/xyz}=\varnothing \&comma;\mathrm{Coordinates/X\_equal\_xyz}=\varnothing \&comma;\mathrm{Coordinates/with\_macros}=\varnothing \right]$

Coordinates(X=spherical);

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\left\{X\right\}$

Remove the alias X to turn visible the information in the next input line

alias(X = X):

CoordinatesINFO();

$\left[\mathrm{Coordinates/X}=\left\{X\right\}\&comma;\mathrm{Coordinates/xyz}=\left\{\left[r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right]\right\}\&comma;\mathrm{Coordinates/X\_equal\_xyz}=\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}\&comma;\mathrm{Coordinates/with\_macros}=\left\{X=\left[\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\right]\right\}\right]$

See Also

Physics[Coordinates], alias, macro

Calling Sequence

Physics:-Library:-Covariant(mu)

Parameters

mu    : a symbol or an algebraic expression

Description

Covariant receives an algebraic expression representing a tensor index, and returns its corresponding covariant index according to: if mu is a symbol prefixed with ~, then remove the ~, otherwise return mu unchanged.

Examples

restart: with(Physics): with(Library):

Covariant(~mu);

$\mathrm{\mu }$

Covariant(mu);

$\mathrm{\mu }$

See Also

Contravariant, FlipCharacter, StripTilde, StripTildeFromRepeatedIndices, ToCovariant

Calling Sequence

Physics:-Library:-Degree()

Parameters

EE  : any Maple algebraic object, or equation or a set or list of them

X   : a name or a set of them

minmax : optional, can be min, max (default), or both, to request the minimal (equivalent to the lower degree), maximal degree with respect to X, or both (returns a sequence of two numbers)

Description

Degree computes the maximum degree of a set of equations EE with respect to variable(s) X, possibly not commutative.

NOTE: if all the equations are equal to 0, the value returned is 0, not -infinity. When the degree cannot be computed, Degree returns FAIL.

Examples

restart: with(Physics): with(Library):

Degree({a, b^2, c^3}, {a, b, c});

$3$

See Also

Physics[Coefficients]

Calling Sequence

Physics:-Library:-FlattenPowersInProduct(A0)

Parameters

A0  : any algebraic object or a table

Description

FlattenPowersInProduct receives an expression, and if it is a power, where the base can be not commutative and the exponent an integer, returns the corresponding product of operands. If A0 is a table, it maps over its elements.

Examples

restart: with(Physics): with(Library):

Setup(op = A);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{op}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{quantumoperators}=\left\{A\right\}\right]$

FlattenPowersInProduct(A^3);

$AAA$

See Also

InertProductToInertPower, PowerToProduct

Calling Sequence

Physics:-Library:-FlipCharacter(mu,{ignorefailure: truefalse := false})

Parameters

mu              : a tensor index, covariant or contravariant

ignorefailure   : optional, when given, if mu is not a covariant/contravariant index, return mu itself instead of FAIL

Description

FlipCharacter flips the covariant or contravariant character of a given index, or returns FAIL when mu is not an index or the covariant/contravariant character cannot be determined.

Examples

restart: with(Physics): with(Library):

Setup(spacetimeindices = greek);

$\left[\mathrm{spacetimeindices}=\mathrm{greek}\right]$

FlipCharacter(mu);

$\mathrm{~\&mu;}$

FlipCharacter(~mu);

$\mathrm{\mu }$

See Also

Contravariant, Covariant, StripTilde, StripTildeFromRepeatedIndices

Calling Sequence

Physics:-Library:-Forget()

Parameters

: no parameters

Description

Forget clears all the remember tables of the internal subroutines of the Physics package, as well as those of simplify, expand, diff, normal, and int. This command is automatically used by all the Physics commands when appropriate, for example, when defining new algebras or tensors, so that computations performed before that with the same mathematical objects, and stored in remember tables, are cleared to permit the next computations take into account the new properties of the objects.

Examples

restart;

expand(A*(B+C));

$AB+AC$

Expand swaps operands in products.

expand((B+C)*A);

$AB+AC$

Expand keeps results in a remember table.

op(4, eval(expand));

$table\left(\left[A\left(B+C\right)=AB+AC\&comma;B+C=B+C\right]\right)$

with(Physics): with(Library):

Setup(op = {A, B, C});

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{op}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$