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Computer Algebra in Theoretical Physics (IOP Webinar)

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Why computer algebra?




... and why computer algebra?

We can concentrate more on the ideas instead of on the algebraic manipulations


We can extend results with ease


We can explore the mathematics surrounding a problem


We can share results in a reproducible way


Representation issues that were preventing the use of computer algebra in Physics



Notation and related mathematical methods that were missing:

coordinate free representations for vectors and vectorial differential operators,

covariant tensors distinguished from contravariant tensors,

functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices

Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics


Inert representations of operations, mathematical functions, and related typesetting were missing:


inert versus active representations for mathematical operations

ability to move from inert to active representations of computations and viceversa as necessary

hand-like style for entering computations and textbook-like notation for displaying results


Key elements of the computational domain of theoretical physics were missing:


ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions

ability to perform computations taking into account custom-defined algebra rules of different kinds

(commutator, anticommutator and bracket rules, etc.)





The Maple computer algebra environment


Classical Mechanics


Inertia tensor for a triatomic molecule


Classical Field Theory


*The field equations for the lambda*Phi^4 model


*Maxwell equations departing from the 4-dimensional Action for Electrodynamics


*The Gross-Pitaevskii field equations for a quantum system of identical particles


Quantum mechanics


*The quantum operator components of  `#mover(mi("L",mathcolor = "olive"),mo("→",fontstyle = "italic"))` satisfy "[L[j],L[k]][-]=i `ε`[j,k,m] L[m]"


Quantization of the energy of a particle in a magnetic field



Show that the energy of a particle in a constant magnetic field oriented along the z axis can be written as


H = `ℏ`*`ω__c`*(`#msup(mi("a",mathcolor = "olive"),mo("†"))`*a+1/2)


where `#msup(mi("a",mathcolor = "olive"),mo("†"))`and a are creation and anihilation operators.





Unitary Operators in Quantum Mechanics


*Eigenvalues of an unitary operator and exponential of Hermitian operators


Properties of unitary operators



Consider two set of kets " | a[n] >" and "| b[n] >", each of them constituting a complete orthonormal basis of the same space.

One can always build an unitary operator U that maps one basis to the other, i.e.: "| b[n] >=U | a[n] >"

*Verify that "U=(&sum;) | b[k] >< a[k] |" implies on  "| b[n] >=U | a[n] >"


*Show that "U=(&sum;) | b[k] > < a[k] | "is unitary


*Show that the matrix elements of U in the "| a[n] >" and  "| b[n] >" basis are equal


Show that A and `&Ascr;` = U*A*`#msup(mi("U"),mo("&dagger;"))`have the same spectrum



Schrödinger equation and unitary transform


Translation operators using Dirac notation


In this section, we focus on the operator T[a] = exp((-I*a*P)*(1/`&hbar;`))



The Action (translation) of the operator T[a]"=(e)^(-i (a P)/(`&hbar;`))" on a ket


Action of T[a] on an operatorV(X)


General Relativity


*Exact Solutions to Einstein's Equations  Lambda*g[mu, nu]+G[mu, nu] = 8*Pi*T[mu, nu]



Main reference: - Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E.  Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. Cambridge University Press, 2003.


The authors reviewed more than 4,000 papers containing solutions to Einstein’s equations in the literature and organized the material into chapters according to the physical properties of these solutions.


These solutions are now digitized within Maple 2016, so that it is now possible to actually compute with them.


The solutions are turned active by a simple call to the g_  spacetime metric.


Everything else gets automatically derived on the fly ( Christoffel symbols  , Ricci  and Riemann  tensors orthonormal and null tetrads , etc.)


Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics  and DifferentialGeometry  packages.


All the mathematics within the Maple library are readily available to work with these solutions.



*"Physical Review D" 87, 044053 (2013)


Given the spacetime metric,

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))

a) Compute the Ricci and Weyl scalars


b) Compute the trace of


"Z[alpha]^(beta)=Phi R[alpha]^(beta)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"


where `&equiv;`(Phi, Phi(r)) is some function of the radial coordinate, R[alpha, `~beta`] is the Ricci tensor, `&Dscr;`[alpha] is the covariant derivative operator and T[alpha, `~beta`] is the stress-energy tensor


T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))

c) Compute the components of "W[alpha]^(beta)"" &equiv;"the traceless part of  "Z[alpha]^(beta)" of item b)


d) Compute an exact solution to the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)" obtained in c)


Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.


a) The Ricci and Weyl scalars


b) The trace of "  Z[alpha]^(beta)=Phi R[alpha]^(beta)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"


b) The components of "W[alpha]^(beta)"" &equiv;"the traceless part of " Z[alpha]^(beta)"


c) An exact solution for the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)"


*The Equivalence problem between two metrics



From the "What is new in Physics in Maple 2016" page:


In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying [Einstein’s] partial differential equations. [By combining that functionality with the one in the Physics and Physics:-Tetrads package] you can also formulate and, depending on the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation.

Example from: A. Karlhede, "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980


*Equivalence for Schwarzschild metric (spherical and Krustal coordinates)



This problem is interesting because:


a) It is well known in the literature

b) It involves departing from a metric expressed in "mixed coordinates"

c) When writting the metric entirely in Krustal coordinates, the dependence involves special functions (LambertW)


Formulation of the problem (remove mixed coordinates)


Solving the Equivalence


Tetrads and Weyl scalars in canonical form



Generally speaking a canonical form is obtained using transformations that leave invariant the tetrad metric in a tetrad system of references, so that theWeyl scalars are fixed as much as possible (conventionally, either equal to 0 or to 1).


Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics.

The implementation is as in "General Relativity, an Einstein century survey", edited by S.W. Hawking (Cambridge) and W. Israel (U. Alberta, Canada), specifically Chapter 7 written by S. Chandrasekhar, page 388:








Residual invariance

Petrov type I







Petrov type II







Petrov type III







Petrov type D






`&Psi;__2`  remains invariant under rotations of Class III

Petrov type N






`&Psi;__4` remains invariant under rotations of Class II



The transformations (rotations of the tetrad system of references) used are of Class I, II and III as defined in Chandrasekar's chapter - equations (7.79) in page 384, (7.83) and (7.84) in page 385. Transformations of Class I can be performed with the command Physics:-Tetrads:-TransformTetrad using the optional argument nullrotationwithfixedl_, of Class II using nullrotationwithfixedn_ and of Class III by calling TransformTetrad(spatialrotationsm_mb_plan, boostsn_l_plane), so with the two optional arguments simultaneously.


The determination of appropriate transformation parameters to be used in these rotations, as well as the sequence of transformations happens all automatically by using the optional argument, canonicalform of TransformTetrad .


restart; with(Physics); with(Tetrads)

`Setting lowercaselatin letters to represent tetrad indices `


0, "%1 is not a command in the %2 package", Tetrads, Physics


0, "%1 is not a command in the %2 package", Tetrads, Physics


[IsTetrad, NullTetrad, OrthonormalTetrad, PetrovType, SimplifyTetrad, TransformTetrad, e_, eta_, gamma_, l_, lambda_, m_, mb_, n_]


Petrov type I


Petrov type II


Petrov type III


Petrov type N


Petrov type D