Physics for Maple 18 - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Physics : Updates : Physics for Maple 18

Physics

 

Maple provides a state-of-the-art environment for algebraic computations in physics, with emphasis on ensuring the computational experience is as natural as possible. The theme of the Physics project for Maple 18 has been the consolidation and integration of the Physics package with the rest of the Maple library, making it even easier to combine standard Maple commands and techniques with Physics-specific computations. With more than 500 enhancements throughout the entire package to increase robustness and versatility, an extension of its typesetting capabilities to support even more standard notation, as well 17 new Physics:-Library commands to support further explorations and extensions, Maple 18 extends the range of physics-related algebraic formulations that can be done in a natural way inside Maple. The impact of these changes is across the board, from vector analysis to quantum mechanics, relativity and field theory.


As part of its commitment to providing the best possible environment for algebraic computations in physics, Maplesoft has launched a Maple Physics: Research and Development web site, where users can download research versions, ask questions, and provide feedback. The results from this accelerated exchange with people around the world have been incorporated into the Physics package in Maple 18.

 

Simplify

4-Vectors, Substituting Tensors

Functional Differentiation

More Metrics in the Database of Solutions to Einstein's Equations

Commutators, AntiCommutators

Expand and Combine

New Enhanced Modes in Physics Setup

Dagger

Vectors Package

Library

Miscellaneous

Simplify

Simplification is perhaps the most common operation performed in a computer algebra system. In Physics, this typically entails simplifying tensorial expressions, or expressions involving noncommutative operators that satisfy certain commutator/anticommutator rules, or sums and integrals involving quantum operators and Dirac delta functions in the summands and integrands. Relevant enhancements were introduced in Maple 18 for all these cases.

Examples

restart; withPhysics: Setupmathematicalnotation = true;

mathematicalnotation=true

(1)

Simplification of sums when the summand is linear in KroneckerDeltas:

Sumsqrtn+1 KroneckerDeltam,n+1+sqrtn KroneckerDeltam,n1,n = 0 .. infinity

n=0n+1δm,n+1+nδm,n1

(2)

Simplify

m+m+1

(3)

Simplification of tensorial expressions. To facilitate typing, set the spacetime indices to be lowercaselatin:

Setupspacetimeindices = lowercaselatin

spacetimeindices=lowercaselatin

(4)

Define a tensor Fa:

DefineFa

Defined objects with tensor properties

γμ,Fa,σμ,μ,gμ,ν,δμ,ν,εα,β,μ,ν

(5)

 

The following tensorial expression,

1010921950548Ff2Fd2Fj2FaFbFc2023081095744Ff2Fd2FaFbFc+1701FcFeFfge,f81FeFf2δc,e1350Feδc,fδe,f81FdFiFjgi,j162Fjδd,iδi,j+108Fjδd,j2511FaFhgd,h+27FdFhga,h+324FdFhδa,h432FbFn2+27Fbgl,n2+6156Fnδb,n81FaFdFk2+81FdFkδa,jδj,k+8100FjFkga,kδd,j81FbFdFi25832FbFdδh,i2+135Fi2δb,d642978Fe2Ffgc,f+8748Ffgc,f+128755884390192Fe2Fh2FaFbFc25470904248Fh2Fn2Fi2FaFbFc148090286178Fe2Fn2Fh2Fi2FaFbFc5022FcFigh,iδd,h1458FdFhgc,iδh,i+81FhFiδc,iδd,h5994FeFggb,eδa,g2700FaFgδb,g9396Fg2ga,b7695FdFkFmδk,m30780Fmδd,kδk,m+500455863936Fd2FaFbFc1850647623120ga,bFh2Fg2Fc18366600960Fe2Fn2Fi2Fd2FaFbFc355087618560Fe2Fd2FaFbFc5179618847700Fh2FaFbFc290804515200Fe2Fd2Fj2FaFbFc462661905780ga,bFh2Fg2Fk2Fc1378008798300Fh2Fn2FaFbFc

1010921950548FdFddFfFffFjFjjFaFbFc2023081095744FdFddFfFffFaFbFc+1701ge,fFcFeeFff81FfFffFeδcece1350Feδc,fδe,fe,f81gi,jFdFiiFjj162Fjδd,iδi,ji,j+108Fjδdjdj2511FaFhhgdhdh+324FhFddδahah+27FddFhhga,h432FnFnnFb+27gl,ngl,nl,nFb+6156Fnδbnbn81FkFkkFaFd+81FdFkδa,jδj,kj,k+8100ga,kFjFkkδdjdj81FbFiFddFii5832FbFddδh,iδh,ih,i+135FiFiiδbdbd642978FeFeegc,fFff+8748gc,fFff+128755884390192FeFeeFhFhhFaFbFc25470904248FhFhhFiFiiFnFnnFaFbFc148090286178FeFeeFhFhhFiFiiFnFnnFaFbFc5022gh,iFcFiiδdhdh1458gc,iFdFhδh,ih,i+81FhFiδciciδdhdh5994gb,eFeeFgδagag2700FaFgδbgbg9396FgFggga,b7695FkFmFddδk,mk,m30780Fmδdkdkδk,mk,m+500455863936FdFddFaFbFc1850647623120FgFggFhFhhga,bFc18366600960FdFddFeFeeFiFiiFnFnnFaFbFc355087618560FdFddFeFeeFaFbFc5179618847700FhFhhFaFbFc290804515200FdFddFeFeeFjFjjFaFbFc462661905780FgFggFhFhhFkFkkga,bFc1378008798300FhFhhFnFnnFaFbFc

(6)

has various terms with contracted indices. In each term, {a,b,c} are free indices:

Check, all

The repeated indices per term are: ...,...,...; the free indices are: ...

d,f,j,d,f,d,e,f,h,i,j,l,n,d,e,f,h,i,j,k,e,h,h,i,n,e,h,i,n,d,e,g,h,i,k,m,d,g,h,d,e,i,n,d,e,h,d,e,j,g,h,k,h,n,a,b,c

(7)

Taking into account Einstein's sum rule for contracted (repeated) indices, the symmetry properties of gi,j and δi,j, this tensorial expression is equal to zero:

Simplify

0

(8)

The simplification of integrals and sums involving quantum operators that satisfy algebra rules is now more powerful, both in the continuous and discrete case. Consider a field, ψ, and its expansion in terms in a basis of functions, φ using operators, a and a, that satisfy:an,ap=δnp

Setupop = a, psi, algebrarules = %Commutatoran, %Daggerap = Diracnp, %Commutatoran, ap = 0

* Partial match of 'op' against keyword 'quantumoperators'

algebrarules=%Commutatoran,ap=0,%Commutatoran,%Daggerap=δnp,quantumoperators=a,ψ

(9)

The expansion of terms ψ and ψis given by:

ψr=φn,ranⅆn

ψr=φn,ranⅆn

(10)

subsn=p,r=s,Dagger

Daggerψs=φp,s&conjugate0;Daggerapⅆp

(11)

The commutator ψr,ψs is equal to:

Commutator,

ψr,Daggerψs=φn,ranⅆn,φp,s&conjugate0;Daggerapⅆp

(12)

expand

ψrDaggerψsDaggerψsψr=φn,ranⅆnφp,s&conjugate0;Daggerapⅆpφp,s&conjugate0;Daggerapⅆpφn,ranⅆn

(13)

The products of integrals on the right-hand side can both be combined into double integrals, then recombined into a single integral and simplified taking into account the algebra rule stated: an,ap= δnp.

Simplify

ψrDaggerψsDaggerψsψr=φp,s&conjugate0;φp,rⅆp

(14)

The step involving only the combination of the integrals can now also be performed separately:

combine

ψrDaggerψsDaggerψsψr=φp,s&conjugate0;φn,rDaggerapan+φn,rφp,s&conjugate0;anDaggerapⅆnⅆp

(15)

The extended capabilities in Simplify regarding integration also work in the discrete case, over sums. Redo the algebra rule now considering the same relations but in the discrete case.

Setupredo, quantumoperators = a, psi, algebrarules=%Commutatoran,%Daggerap=KroneckerDeltan,p,%Commutatoran,ap=0, spacetimeindices = greek

algebrarules=%Commutatoran,ap=0,%Commutatoran,%Daggerap=δn,p,quantumoperators=a,ψ,spacetimeindices=greek

(16)


The following sum can now be simplified by combining the sums and taking into account the new (discrete) algebra rules, or just performing the combination step:

ψr=Sumφnran,n=..

ψr=n=φnran

(17)

subsn=p,r=s,Dagger

Daggerψs=p=φps&conjugate0;Daggerap

(18)

expandCommutator,

ψrDaggerψsDaggerψsψr=n=φnranp=φps&conjugate0;Daggerapp=φps&conjugate0;Daggerapn=φnran

(19)

Simplify

ψrDaggerψsDaggerψsψr=p=φps&conjugate0;φpr

(20)

combine

ψrDaggerψsDaggerψsψr=p=n=φps&conjugate0;φnrDaggerapan+φnrφps&conjugate0;anDaggerap

(21)

Improvements in the simplification of annihilation and the creation of fermionic operators, as well as the related occupation number operator:

Setupanticommutativeprefix = psi

anticommutativeprefix=_λ,ψ

(22)

am  Annihilationpsi, notation = explicit

amaψ1

(23)

ap  Creationpsi, notation = explicit

apa+ψ1

(24)

The related occupation number operator:

N  ap . am

Na+ψ1aψ1

(25)

Consider the application of these fermionic operators to a related state vector:

Ketpsi, 1

Ketψ,1

(26)

am . 

Ketψ,0

(27)

am . 

0

(28)

Increasing the occupation number,

ap . 

Ketψ,1

(29)

ap . 

0

(30)

In other words, powers of annihilation and creation fermionic operators are equal to zero:

am2

aψ12

(31)

Simplifyam2

0

(32)

Simplifyap2

0

(33)

The occupation number operator is also idempotent:

N N N = N

a+ψ1aψ1a+ψ1aψ1a+ψ1aψ1=a+ψ1aψ1

(34)

These expressions can now be simplified:

Simplify%

a+ψ1aψ1=a+ψ1aψ1

(35)

The simplification of vectorial expressions was also enhanced. For example:

withVectors :

B0·v&xB0+B1

B0·v×B0+B1

(36)

Simplify

B0·v×B1

(37)

4-Vectors, Substituting Tensors

In Maple 17, it is possible to define a tensor with a tensorial equation, where the tensor being defined is on the left-hand side. Then, on the right-hand side, you write either a tensorial expression with free and repeated indices, or a Matrix or Array with the components themselves. In Maple 18, you can also define a 4-Vector with a tensorial equation, where you indicate the vector's components on the right-hand side as a list.

One new Library routine specialized for tensor substitutions was added to the Maple library: SubstituteTensor, which substitutes the equation(s) Eqs into an expression, taking care of the free and repeated indices, such that: 1) equations in Eqs are interpreted as mappings having the free indices as parameters, and 2) repeated indices in Eqs do not clash with repeated indices in the expression. This new routine can also substitute algebraic sub-expressions of type product or sum within the expression, generalizing and unifying the functionality of the subs and algsubs commands for algebraic tensor expressions.

Examples

restart; withPhysics: Setupmathematicalnotation = true;

mathematicalnotation=true

(38)

Define a contravariant 4-vector with components p__x, p__y, p__z,p__t:

Definep~mu = p__x, p__y, p__z,p__t

Defined objects with tensor properties

γμ,σμ,μ,gμ,ν,pμμ,δμ,ν,εα,β,μ,ν

(39)

You can retrieve the components in different ways:

p~mu = Library:-TensorComponentsp~mu

pμμ=p__x,p__y,p__z,p__t

(40)

pmu = Library:-TensorComponentspmu

pμ=p__x,p__y,p__z,p__t

(41)

Or, indexing p with a contravariant or covariant integer value of the index:

p~1p1

p__xp__x

(42)

You can compute with pμμ algebraically; p[~mu] will return its components only when the index assumes integer values 0μ 4.

pmu pnu ep_mu,nu,alpha,beta

εα,β,μ,νpμμpνν

(43)

Simplify

0

(44)

pmu2

pμpμμ

(45)

SumOverRepeatedIndices

p__t2p__x2p__y2p__z2

(46)

Define some tensors for experimentation with the new Library:-SubstituteTensor command:

DefineA,B,F,G

Defined objects with tensor properties

A,B,F,G,γμ,σμ,μ,gμ,ν,pμμ,δμ,ν,εα,β,μ,ν

(47)

A substitution equation:

Aμ=Gν,αAαFμ,ν

Aμ=Gν,αAααFμνμν

(48)

Substitute into AνAνν: the free indices of (48) are taken as parameters, repeated indices in the substitution equation do not repeat more than once in the result:

Library:-SubstituteTensor, AνAνν

Gβ,αAααFνβνβGλ,κAκκFν,λν,λ

(49)

When the left-hand side of the substitution equation is a tensor function, the functionality is also taken as a parameter,

Library:-SubstituteTensorAμX=BμX,AνY

BνY

(50)

SubstituteTensor can also substitute sub-expressions of type product or sum, similar to what algsubs does, so for example substitute:

Amu Bnu = Gmu,nu

AμBν=Gμ,ν

(51)

into,

Aalpha Fmu,nu Bbeta Arho Brho Gmu,nu

AαFμ,νGμ,νμ,νBβAρBρρ

(52)

Library:-SubstituteTensor,

Gα,βGρρρρFμ,νGμ,νμ,ν

(53)

Check the free and repeated indices of this result and verify that the free indices of (52) are the same:

Check,all

The repeated indices per term are: ...,...,...; the free indices are: ...

μ,ν,ρ,α,β

(54)

Check,free

The free indices are: ...

α,β

(55)

Functional Differentiation

The Physics:-Fundiff command for functional differentiation has been extended to handle all the complex components (abs, argument, conjugate, Im, Re, signum) and vectorial differential operators in order to compute field equations using variational principles when the field function enters the Lagrangian together with its conjugate. For an example illustrating the use of the new capabilities in the context of a more general problem, see the MaplePrimes post Quantum Mechanics using Computer Algebra.

Examples

restart; withPhysics:Setupmathematicalnotation = true

mathematicalnotation=true

(56)

A function and its conjugate are considered independent from each other regarding functional differentiation:

%Fundiff = Fundiffconjugatefx, fy

%Fundifffx&conjugate0;,fy=0

(57)

Fundiff can now compute functional derivatives of expressions involving vectorial differential operators and the corresponding conjugate functions.

withVectors:

%Fundiff = Fundiff%Gradientfx, fy

%Fundiff%Gradientfx,fy=δxyi

(58)

Set a system of coordinates for functional differentiation with many variables:

CoordinatesQ = X, Y, Z, T

Default differentiation variables for d_, D_ and dAlembertian are: Q=X,Y,Z,T

Systems of spacetime Coordinates are: Q=X,Y,Z,T

Q

(59)

The Action for a system:

S  IntcNorm%GradientPhix,y,z,t2, x,y,z,t

SNorm%GradientΦx,y,z,t2ⅆxⅆyⅆzⅆt

(60)

The equations of motion through functional differentiation:

%Fundiff = FundiffS, PhiX,Y,Z,T

%FundiffNorm%GradientΦx,y,z,t2ⅆxⅆyⅆzⅆt,ΦQ=2Z2ΦQ&conjugate0;2Y2ΦQ&conjugate0;2X2ΦQ&conjugate0;

(61)

The Action for a complex scalar field with a Lagrangian quadratic in the derivatives: note that abs now automatically maps into the Norm of the vector.

SIntc12absGradientΦx,y,z,t2, x,y,z,t

SxΦx,y,z,t&conjugate0;xΦx,y,z,t2yΦx,y,z,t&conjugate0;yΦx,y,z,t2zΦx,y,z,t&conjugate0;zΦx,y,z,t2ⅆxⅆyⅆzⅆt

(62)

The corresponding field equation:

FundiffS, ΦX,Y,Z,T  = 0

2Z2ΦQ&conjugate0;2+2Y2ΦQ&conjugate0;2+2X2ΦQ&conjugate0;2=0

(63)

More Metrics in the Database of Solutions to Einstein's Equations

A database of solutions to Einstein's equations was added to the Maple library in Maple 15 with a selection of metrics from "Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E.,  Exact Solutions to Einstein's Field Equations" and "Hawking, Stephen; and Ellis, G. F. R., The Large Scale Structure of Space-Time". More metrics from these two books were added for Maple 16 and Maple 17. These metrics can be searched using the command DifferentialGeometry:-Library:-MetricSearch, or directly using g_ (the Physics command representing the spacetime metric that also sets the metric to your choice).

• 

For Maple 18, fifty more metrics were added to the database from Chapter 28 of the aforementioned book entitled "Exact Solutions to Einstein's Field Equations".

• 

It is now possible to list all the metrics of a chapter by indexing the metric command with the chapter's number, for example, entering g_["28"].

Examples

restart; withPhysics:Setupmathematicalnotation = true

mathematicalnotation=true

(64)

By default, the metric is a Minkowski type:

g_

You can query about metrics directly from the metric command g_

g_Bajer

____________________________________________________________

28,58.2,1=Authors=Bajer, Kowalezynski (1985),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=RobinsonTrautman,Comments=All tensor components given with respect to the anholonomic frame,The coordinates xi and xi1 are complex conjugates

____________________________________________________________

28,58.3,1=Authors=Bajer, Kowalezynski (1985),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=RobinsonTrautman,Stationary,Comments=Case 1 of 2,The coordinates xi and xi1 are complex conjugates, the parameters _Q0 and _Q01 are complex conjugates,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,58.3,2=Authors=Bajer, Kowalezynski (1985),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=RobinsonTrautman,Comments=Case 1 of 2,The coordinates xi and xi1 are complex conjugates, the parameters _Q0 and _Q01 are complex conjugates,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,58.4,1=Authors=Bajer, Kowalezynski (1985),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=RobinsonTrautman,Comments=The coordinates xi and xi1 are complex conjugates,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

Warning, found more than one match for the keyword 'Bajer', as seen above. Please refine your 'keyword' or re-enter the metric 'g_[...]' with the list of three numbers identifying the metric, for example as in g_[[28, 58.2, 1]] or Setup(metric = [28, 58.2, 1])

When you identified the metric, you can set it directly from g_ (alternatively, you can do that using Setup):

g_28, 58.2, 1

Systems of spacetime Coordinates are: X=r,ξ,ξ1,u

Default differentiation variables for d_, D_ and dAlembertian are: X=r,ξ,ξ1,u

The Bajer, Kowalezynski (1985) metric in coordinates r,ξ,ξ1,u

Parameters: κ0,Q0,Q01,m0,b,Q

Comments: All tⅇnsor componⅇnts gⅈvⅇn wⅈth rⅇspⅇct to thⅇ anholonomⅈc framⅇ

New in Maple 18, you can now also list all the metrics of a chapter. For example, for the metrics of Chapter 28,

g_28

____________________________________________________________

28,16,1=Authors=Robinson-Trautman (1962),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=The coordinate zeta is changed to xi,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,17,1=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=The coordinate zeta is changed to xi,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence

____________________________________________________________

28,21,1=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=The coordinate zeta is changed to xi

____________________________________________________________

28,21,2=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=The coordinate zeta is changed to xi

____________________________________________________________

28,21,3=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Static,Comments=The coordinate zeta is changed to xi

____________________________________________________________

28,21,4=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=The coordinate zeta is changed to xi,This is _a special case of Kasner spacetime Stephani [13, 51,1], [13, 53,1]

____________________________________________________________

28,21,5=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Static,Comments=The coordinate zeta is changed to xi

____________________________________________________________

28,21,6=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=The coordinate zeta is changed to xi

____________________________________________________________

28,21,7=Authors=Robinson (1975),Foster, Newman (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Static,Comments=The coordinate zeta is changed to xi

____________________________________________________________

28,24,1=Authors=Collinson, French (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=Stephani claims this metric is static which is false since the orbit type is generically Riemannian,The assumption _b >0 and _c >0 is made so that the given base point is in the domain

____________________________________________________________

28,25,1=Authors=Collinson, French (1967),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,26,1=Authors=Robinson, Trautman (1962),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=One can use the diffeo r -> -r and u -> -u to make the assumption r > 0,The case _m = 0 is Stephani, [28, 16,1],The metric is type D at points where r = 3*_m/(xi1+xi2) and type II on either side of this hypersurface. For convenience, it is assumed that 3*_m - r*(xi1 + xi2) > 0,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,26,2=Authors=Robinson, Trautman (1962),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=One can use the diffeo r -> -r and u -> -u to make the assumption r > 0,The case _m = 0 is Stephani, [28, 16,1].,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,26,3=Authors=Robinson, Trautman (1962),PrimaryDescription=Vacuum,SecondaryDescription=RobinsonTrautman,Comments=One can use the diffeo r -> -r and u -> -u to make the assumption r > 0,The case _m = 0 is Stephani, [28, 16,1].,AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,41,1=Authors= Bartrum (1967),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=PureRadiation,RobinsonTrautman,Comments=AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28,43,1=Authors=Robinson, Trautman (1962),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=PureRadiation,RobinsonTrautman,Comments=h1(u) is the conjugate of h(u)

____________________________________________________________

28&comma;44&comma;1=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 1 of 6. K = 1, Riemmannian Orbits&comma;_Q1 is the conjugate of Q&comma;The metric is defined for all r > 0. The restriction 2*r^2 - 4*_m*r +_kappa0*_Q*_Q1 < 0 gives Riemannian orbits for the isometry group

____________________________________________________________

28&comma;44&comma;2=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Static&comma;Comments=Case 2 of 6 K = 1, PseudoRiemmannian Orbits&comma;_Q1 is the conjugate of Q&comma;The metric is defined for all r > 0. The restriction 2*r^2 - 4*_m*r +_kappa0*_Q*_Q1 > 0 gives PseudoRiemannian orbits for the isometry group

____________________________________________________________

28&comma;44&comma;3=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 3 of 6 K = 0, Riemmannian Orbits&comma;_Q1 is the conjugate of Q&comma;The metric is defined for all r > 0. The restriction _Q*_Q1*_kappa0-4*_m*r < 0 gives Riemannian orbits for the isometry group

____________________________________________________________

28&comma;44&comma;4=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Static&comma;Comments=Case 4 of 6 K = 0, Pseudo-Riemmannian Orbits&comma;_Q1 is the conjugate of Q&comma;The metric is defined for all r > 0. The restriction _Q*_Q1*_kappa0-4*_m*r > 0 gives Pseudo-Riemannian orbits for the isometry group

____________________________________________________________

28&comma;44&comma;5=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case of 6. K = 1, Riemmannian Orbits&comma;_Q1 is the conjugate of _Q&comma;The metric is defined for all r > 0. The restriction _Q*_Q1*_kappa0 - 4*m*r- 2*r^2 < 0 gives Riemannian orbits for the isometry group

____________________________________________________________

28&comma;44&comma;6=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Static&comma;Comments=Case 6 of 6. K = 1, Pseudo-Riemmannian Orbits&comma;_Q1 is the conjugate of _Q&comma;The metric is defined for all r > 0. The restriction 2*r^2 - 4*_m*r +_kappa0*_Q*_Q1 > 0 gives Pseudo-Riemannian orbits for the isometry group

____________________________________________________________

28&comma;45&comma;1=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=Static&comma;Comments=Case 1 of 2&comma;Note that the metric only depends on the square of the function P. If the funtion P^(-2) is negative, then the solution is static.&comma;The parameter _q determines _a duality rotation on the electromagnetic field.

____________________________________________________________

28&comma;45&comma;2=Authors=Leroy (1976)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=&comma;Comments=Case 2 of 2&comma;Note that the metric only depends on the square of the function P. If the funtion P^(-2) is negative, then the solution is static.&comma;The parameter _q determines _a duality rotation on the electromagnetic field.

____________________________________________________________

28&comma;46&comma;1=PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 1 of 2

____________________________________________________________

28&comma;46&comma;2=PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Stationary&comma;Comments=Case 2 of 2

____________________________________________________________

28&comma;53&comma;1=Authors=Bartrum (1967)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;PureRadiation&comma;Stationary&comma;Comments=Case 1 of 2. We take _m > 0 for simplicity. AlternativeNullTetrad1 is the standard Robinson Trautman null tetrad. The orbit type is pseudo-Riemannian at the given base point if _f1(0)^2*_f2(0)^2 > 2_m. If alpha = constant then the electromagnetic field is inheriting.

____________________________________________________________

28&comma;53&comma;2=Authors=Bartrum (1967)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;PureRadiation&comma;Comments=Case 2 of 2. We take _m > 0 for simplicity. AlternativeNullTetrad1 is the standard Robinson Trautman null tetrad. The orbit type is pseudo-Riemannian at the given base point if _f1(0)^2*_f2(0)^2 < 2_m. If alpha = constant then the electromagnetic field is inheriting.

____________________________________________________________

28&comma;55&comma;1=PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Stationary&comma;Comments=Case 1 of 2. We choose _A > 0 and u > 0 for simplicity. AlternativeNullTetrad1 is the standard Robinson Trautman null tetrad.

____________________________________________________________

28&comma;55&comma;2=PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 2 of 2. We choose _A > 0 and u > 0 for simplicity. AlternativeNullTetrad1 is the standard Robinson Trautman null tetrad.

____________________________________________________________

28&comma;60&comma;1=Authors=Kowalczynski (1978)&comma;Kowalczynski (1985)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates, the parameters _Q0 and _Q01 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;61&comma;1=Authors=Kowalczynski (1978)&comma;Kowalczynski (1985)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates, the parameters _Q0 and _Q01 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;64&comma;1=Authors=Herlt and Stephani (1984)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;66&comma;1=Authors=Herlt and Stephani (1984)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;67&comma;1=Authors=Herlt and Stephani (1984)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;68&comma;1=Authors=Herlt and Stephani (1984)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;72&comma;1=PrimaryDescription=PureRadiation&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;73&comma;1=Authors=Frolov and Khlebnikov (1975)&comma;PrimaryDescription=PureRadiation&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates

____________________________________________________________

28&comma;74&comma;1=Authors=Frolov and Khlebnikov (1975)&comma;PrimaryDescription=PureRadiation&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=With _m(u) = constant, the metric is Ricci flat and becomes 28.24 in Stephani.

____________________________________________________________

28&comma;56.1&comma;1=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Stationary&comma;Comments=Case 1 of 3, epsilon = 0, stationary&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;56.2&comma;2=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 2 of 3, epsilon = 0&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;56.2&comma;3=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 3 of 3, epsilon = 1&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;56.3&comma;1=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;56.4&comma;1=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;56.5&comma;1=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;56.6&comma;1=Authors=Leroy, 1976&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;58.2&comma;1=Authors=Bajer, Kowalezynski (1985)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=All tensor components given with respect to the anholonomic frame&comma;The coordinates xi and xi1 are complex conjugates

____________________________________________________________

28&comma;58.3&comma;1=Authors=Bajer, Kowalezynski (1985)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Stationary&comma;Comments=Case 1 of 2&comma;The coordinates xi and xi1 are complex conjugates, the parameters _Q0 and _Q01 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;58.3&comma;2=Authors=Bajer, Kowalezynski (1985)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=Case 1 of 2&comma;The coordinates xi and xi1 are complex conjugates, the parameters _Q0 and _Q01 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;58.4&comma;1=Authors=Bajer, Kowalezynski (1985)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinates xi and xi1 are complex conjugates&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

Warning, found more than one match for the keyword '28', as seen above. Please refine your 'keyword' or re-enter the metric 'g_[...]' with the list of three numbers identifying the metric, for example as in g_[[28, 16, 1]] or Setup(metric = [28, 16, 1])

Commutators, AntiCommutators

When computing with products of noncommutative operators, the results depend on the algebra of commutators and anticommutators that you previously set. Besides that, in Physics, various mathematical objects themselves satisfy specific commutation rules. You can query about these rules using the Library commands Commute and Anticommute. Previously existing functionality and enhancements in this area were refined and implemented in Maple 18. Among them:

• 

Both Commutator and AntiCommutator now accept matrices as arguments.

• 

The AntiCommutator of products of fermionic operators - for instance annihilation and creation operators - is now derived automatically from the intrinsic anticommutation rules they satisfy.

• 

Commutators and Anticommutators of vectorial quantum operators A,B, are now implemented and expressed using the dot (scalar) product, as in A,B=A·BB·A

• 

If two noncommutative operators a and S  satisfy a&comma;S=0 , then the commutator  a&comma;S is automatically taken equal to 0; if in addition S is Hermitian, then  a&comma;Sis also automatically taken equal to zero.

Examples

restart&semi; withPhysics&colon;Setupmathematicalnotation &equals; true

mathematicalnotation=true

(65)

Commutator and AntiCommutator now also operate on matrices:

M__1 Matrix2&comma;2&comma;a&comma;b&comma;c&comma;d

M__1abcd

(66)

M__2 Matrix2&comma;2&comma;alpha&comma;beta&comma;gamma&comma;delta

M__2αβγδ

(67)

%Commutator &equals; CommutatorM__1&comma; M__2

%Commutatorabcd&comma;αβγδ=bγcβaβαb+bδβdγa+cαδc+dγbγ+cβ

(68)

Commutators of vectorial operators were implemented, expressed using the scalar (dot) product:

withVectors&colon; Setupop &equals; A_&comma; B_

* Partial match of 'op' against keyword 'quantumoperators'

quantumoperators=A&comma;B

(69)

Commutator &equals; expand&commat;CommutatorA_&comma; B_

A,B=A·BB·A

(70)

Define 4 pairs of annihilation/creation operators for fermionic particles:

Setupanticommutativeprefix &equals; psi

anticommutativeprefix=_&lambda;&comma;ψ

(71)

for j to 4 do      apj  Creationpsi&comma; j&comma; notation &equals; explicit&semi;      amj  Annihilationpsi&comma; j&comma; notation &equals; explicit  end do&semi;

ap1a+ψ1

am1aψ1

ap2a+ψ2

am2aψ2

ap3a+ψ3

am3aψ3

ap4a+ψ4

am4aψ4

(72)

For these operators, the system knows about the anticommutator rules satisfied between any two of them, the algebra is set on the fly when you define them.

Setupalgebra

* Partial match of 'algebra' against keyword 'algebrarules'

algebrarules=%AntiCommutatoraψ1&comma;a+ψ1=1&comma;%AntiCommutatoraψ2&comma;a+ψ2=1&comma;%AntiCommutatoraψ3&comma;a+ψ3=1&comma;%AntiCommutatoraψ4&comma;a+ψ4=1

(73)

Using that information, the system now also knows about the anticommutator of products of these fermionic operators. For example, on the left-hand side: inert, on the right-hand side: computed.

%AntiCommutator &equals; AntiCommutatorap1&comma;am1am2ap2

%AntiCommutatora+ψ1&comma;aψ1aψ2a+ψ2=aψ2a+ψ2

(74)

This new functionality is automatically used when sorting products of non-commutative operators according to a specified ordering using the new Library:-SortProducts routine; let P be a product:

P  ap2 am2 am1 ap1

Pa+ψ2aψ2aψ1a+ψ1

(75)

Rewrite this product using the following ordering: with the creation operators to the left, using the anticommutator relations between them.

ApAm  seqapj&comma; j &equals; 1 .. 2&comma; seqamj&comma; j &equals; 1 .. 2

ApAma+ψ1&comma;a+ψ2&comma;aψ1&comma;aψ2

(76)

Library:-SortProductsP&comma; ApAm&comma; useanticommutator

a+ψ1a<