Physics for Maple 18

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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

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

>	$restart &semi; with (Physics) &colon; Setup (mathematicalnotation = true) &semi;$

$[mathematicalnotation = true]$

(1)

Simplification of sums when the summand is linear in KroneckerDeltas:

>	$Sum (sqrt (n + 1) KroneckerDelta [m, n + 1] + sqrt (n) KroneckerDelta [m, n - 1], n = 0 .. infinity)$

$\sum_{n = 0}^{\infty} (\sqrt{n + 1} δ_{m, n + 1} + \sqrt{n} δ_{m, n - 1})$

(2)

>	$Simplify ()$

$\sqrt{m} + \sqrt{m + 1}$

(3)

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

>	$Setup (spacetimeindices = lowercaselatin)$

$[spacetimeindices = lowercaselatin]$

(4)

Define a tensor $F_{a} &colon;$

>	$Define (F [a])$

$Defined objects with tensor properties$

$\{γ_{μ}, F_{a}, σ_{μ}, \partial_{μ}, g_{μ, ν}, δ_{μ, ν}, ε_{α, β, μ, ν}\}$

(5)

The following tensorial expression,

$- 1010921950548 F_{f}^{2} F_{d}^{2} F_{j}^{2} F_{a} F_{b} F_{c} - 2023081095744 F_{f}^{2} F_{d}^{2} F_{a} F_{b} F_{c} + (1701 F_{c} F_{e} F_{f} g_{e, f} - 81 F_{e} F_{f}^{2} δ_{c, e} - 1350 F_{e} δ_{c, f} δ_{e, f}) (81 F_{d} F_{i} F_{j} g_{i, j} - 162 F_{j} δ_{d, i} δ_{i, j} + 108 F_{j} δ_{d, j}) (2511 F_{a} F_{h} g_{d, h} + 27 F_{d} F_{h} g_{a, h} + 324 F_{d} F_{h} δ_{a, h}) (432 F_{b} F_{n}^{2} + 27 F_{b} g_{l, n}^{2} + 6156 F_{n} δ_{b, n}) - (81 F_{a} F_{d} F_{k}^{2} + 81 F_{d} F_{k} δ_{a, j} δ_{j, k} + 8100 F_{j} F_{k} g_{a, k} δ_{d, j}) (81 F_{b} F_{d} F_{i}^{2} - 5832 F_{b} F_{d} δ_{h, i}^{2} + 135 F_{i}^{2} δ_{b, d}) (- 642978 F_{e}^{2} F_{f} g_{c, f} + 8748 F_{f} g_{c, f}) + 128755884390192 F_{e}^{2} F_{h}^{2} F_{a} F_{b} F_{c} - 25470904248 F_{h}^{2} F_{n}^{2} F_{i}^{2} F_{a} F_{b} F_{c} - 148090286178 F_{e}^{2} F_{n}^{2} F_{h}^{2} F_{i}^{2} F_{a} F_{b} F_{c} - (- 5022 F_{c} F_{i} g_{h, i} δ_{d, h} - 1458 F_{d} F_{h} g_{c, i} δ_{h, i} + 81 F_{h} F_{i} δ_{c, i} δ_{d, h}) (- 5994 F_{e} F_{g} g_{b, e} δ_{a, g} - 2700 F_{a} F_{g} δ_{b, g} - 9396 F_{g}^{2} g_{a, b}) (- 7695 F_{d} F_{k} F_{m} δ_{k, m} - 30780 F_{m} δ_{d, k} δ_{k, m}) + 500455863936 F_{d}^{2} F_{a} F_{b} F_{c} - 1850647623120 g_{a, b} F_{h}^{2} F_{g}^{2} F_{c} - 18366600960 F_{e}^{2} F_{n}^{2} F_{i}^{2} F_{d}^{2} F_{a} F_{b} F_{c} - 355087618560 F_{e}^{2} F_{d}^{2} F_{a} F_{b} F_{c} - 5179618847700 F_{h}^{2} F_{a} F_{b} F_{c} - 290804515200 F_{e}^{2} F_{d}^{2} F_{j}^{2} F_{a} F_{b} F_{c} - 462661905780 g_{a, b} F_{h}^{2} F_{g}^{2} F_{k}^{2} F_{c} - 1378008798300 F_{h}^{2} F_{n}^{2} F_{a} F_{b} F_{c}$

$- 1010921950548 F_{d} F_{}^{d} F_{f} F_{}^{f} F_{j} F_{}^{j} F_{a} F_{b} F_{c} - 2023081095744 F_{d} F_{}^{d} F_{f} F_{}^{f} F_{a} F_{b} F_{c} + (1701 g_{e, f} F_{c} F_{}^{e} F_{}^{f} - 81 F_{f} F_{}^{f} F_{e} δ_{c}^{e} - 1350 F_{e} δ_{c, f} δ_{}^{e, f}) (81 g_{i, j} F_{d} F_{}^{i} F_{}^{j} - 162 F_{j} δ_{d, i} δ_{}^{i, j} + 108 F_{j} δ_{d}^{j}) (2511 F_{a} F_{}^{h} g_{h}^{d} + 324 F_{h} F_{}^{d} δ_{a}^{h} + 27 F_{}^{d} F_{}^{h} g_{a, h}) (432 F_{n} F_{}^{n} F_{b} + 27 g_{l, n} g_{}^{l, n} F_{b} + 6156 F_{n} δ_{b}^{n}) - (81 F_{k} F_{}^{k} F_{a} F_{d} + 81 F_{d} F_{k} δ_{a, j} δ_{}^{j, k} + 8100 g_{a, k} F_{j} F_{}^{k} δ_{d}^{j}) (81 F_{b} F_{i} F_{}^{d} F_{}^{i} - 5832 F_{b} F_{}^{d} δ_{h, i} δ_{}^{h, i} + 135 F_{i} F_{}^{i} δ_{b}^{d}) (- 642978 F_{e} F_{}^{e} g_{c, f} F_{}^{f} + 8748 g_{c, f} F_{}^{f}) + 128755884390192 F_{e} F_{}^{e} F_{h} F_{}^{h} F_{a} F_{b} F_{c} - 25470904248 F_{h} F_{}^{h} F_{i} F_{}^{i} F_{n} F_{}^{n} F_{a} F_{b} F_{c} - 148090286178 F_{e} F_{}^{e} F_{h} F_{}^{h} F_{i} F_{}^{i} F_{n} F_{}^{n} F_{a} F_{b} F_{c} - (- 5022 g_{h, i} F_{c} F_{}^{i} δ_{d}^{h} - 1458 g_{c, i} F_{d} F_{h} δ_{}^{h, i} + 81 F_{h} F_{i} δ_{c}^{i} δ_{d}^{h}) (- 5994 g_{b, e} F_{}^{e} F_{g} δ_{a}^{g} - 2700 F_{a} F_{g} δ_{b}^{g} - 9396 F_{g} F_{}^{g} g_{a, b}) (- 7695 F_{k} F_{m} F_{}^{d} δ_{}^{k, m} - 30780 F_{m} δ_{k}^{d} δ_{}^{k, m}) + 500455863936 F_{d} F_{}^{d} F_{a} F_{b} F_{c} - 1850647623120 F_{g} F_{}^{g} F_{h} F_{}^{h} g_{a, b} F_{c} - 18366600960 F_{d} F_{}^{d} F_{e} F_{}^{e} F_{i} F_{}^{i} F_{n} F_{}^{n} F_{a} F_{b} F_{c} - 355087618560 F_{d} F_{}^{d} F_{e} F_{}^{e} F_{a} F_{b} F_{c} - 5179618847700 F_{h} F_{}^{h} F_{a} F_{b} F_{c} - 290804515200 F_{d} F_{}^{d} F_{e} F_{}^{e} F_{j} F_{}^{j} F_{a} F_{b} F_{c} - 462661905780 F_{g} F_{}^{g} F_{h} F_{}^{h} F_{k} F_{}^{k} g_{a, b} F_{c} - 1378008798300 F_{h} F_{}^{h} F_{n} F_{}^{n} F_{a} F_{b} F_{c}$

(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 $g_{i, j}$ and $δ_{i, j}$ , 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: ${[a (n), {a (p)}^{†}]}_{-} = δ (n - p)$

>	$Setup (op = \{a, psi\}, algebrarules = \{%Commutator (a (n), %Dagger (a (p))) = Dirac (n - p), %Commutator (a (n), a (p)) = 0\})$

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

$[algebrarules = \{{[a (n), a (p)]}_{-} = 0, {[a (n), {a (p)}^{†}]}_{-} = δ (n - p)\}, quantumoperators = \{a, ψ\}]$

(9)

The expansion of terms $ψ$ and $ψ^{†}$ is given by:

>	$ψ (r) = \int_{- \infty}^{\infty} φ (n, r) a (n) &DifferentialD; n$

$ψ (r) = \int_{- \infty}^{\infty} φ (n, r) a (n) &DifferentialD; n$

(10)

>	$subs (\{n = p, r = s\}, Dagger ())$

${ψ (s)}^{†} = \int_{- \infty}^{\infty} \overset{&conjugate0;}{φ (p, s)} {a (p)}^{†} &DifferentialD; p$

(11)

The commutator ${[ψ (r), ψ^{†} (s)]}_{}$ is equal to:

>	$Commutator (,)$

${[ψ (r), {ψ (s)}^{†}]}_{-} = {[\int_{- \infty}^{\infty} φ (n, r) a (n) &DifferentialD; n, \int_{- \infty}^{\infty} \overset{&conjugate0;}{φ (p, s)} {a (p)}^{†} &DifferentialD; p]}_{-}$

(12)

>	$expand ()$

$ψ (r) {ψ (s)}^{†} - {ψ (s)}^{†} ψ (r) = (\int_{- \infty}^{\infty} φ (n, r) a (n) &DifferentialD; n) (\int_{- \infty}^{\infty} \overset{&conjugate0;}{φ (p, s)} {a (p)}^{†} &DifferentialD; p) - (\int_{- \infty}^{\infty} \overset{&conjugate0;}{φ (p, s)} {a (p)}^{†} &DifferentialD; p) (\int_{- \infty}^{\infty} φ (n, r) a (n) &DifferentialD; n)$

(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: ${[a (n), {a (p)}^{†}]}_{-} = δ (n - p) &period;$

>	$Simplify ()$

$ψ (r) {ψ (s)}^{†} - {ψ (s)}^{†} ψ (r) = \int_{- \infty}^{\infty} \overset{&conjugate0;}{φ (p, s)} φ (p, r) &DifferentialD; p$

(14)

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

>	$combine ()$

$ψ (r) {ψ (s)}^{†} - {ψ (s)}^{†} ψ (r) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (- \overset{&conjugate0;}{φ (p, s)} φ (n, r) {a (p)}^{†} a (n) + φ (n, r) \overset{&conjugate0;}{φ (p, s)} a (n) {a (p)}^{†}) &DifferentialD; n &DifferentialD; p$

(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.

>	$Setup (redo, quantumoperators = \{a, psi\}, algebrarules = \{%Commutator (a_{n}, %Dagger (a_{p})) = KroneckerDelta [n, p], %Commutator (a_{n}, a_{p}) = 0\}, spacetimeindices = greek)$

$[algebrarules = \{{[a_{n}, a_{p}]}_{-} = 0, {[a_{n}, {a_{p}}^{†}]}_{-} = δ_{n, p}\}, quantumoperators = \{a, ψ\}, spacetimeindices = greek]$

(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:

>	$ψ (r) = Sum (φ_{n} (r) a_{n}, n = - \infty .. \infty)$

$ψ (r) = \sum_{n = - \infty}^{\infty} φ_{n} (r) a_{n}$

(17)

>	$subs (\{n = p, r = s\}, Dagger ())$

${ψ (s)}^{†} = \sum_{p = - \infty}^{\infty} \overset{&conjugate0;}{φ_{p} (s)} {a_{p}}^{†}$

(18)

>	$expand (Commutator (,))$

$ψ (r) {ψ (s)}^{†} - {ψ (s)}^{†} ψ (r) = (\sum_{n = - \infty}^{\infty} φ_{n} (r) a_{n}) (\sum_{p = - \infty}^{\infty} \overset{&conjugate0;}{φ_{p} (s)} {a_{p}}^{†}) - (\sum_{p = - \infty}^{\infty} \overset{&conjugate0;}{φ_{p} (s)} {a_{p}}^{†}) (\sum_{n = - \infty}^{\infty} φ_{n} (r) a_{n})$

(19)

>	$Simplify ()$

$ψ (r) {ψ (s)}^{†} - {ψ (s)}^{†} ψ (r) = \sum_{p = - \infty}^{\infty} \overset{&conjugate0;}{φ_{p} (s)} φ_{p} (r)$

(20)

>	$combine ()$

$ψ (r) {ψ (s)}^{†} - {ψ (s)}^{†} ψ (r) = \sum_{p = - \infty}^{\infty} \sum_{n = - \infty}^{\infty} (- \overset{&conjugate0;}{φ_{p} (s)} φ_{n} (r) {a_{p}}^{†} a_{n} + φ_{n} (r) \overset{&conjugate0;}{φ_{p} (s)} a_{n} {a_{p}}^{†})$

(21)

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

>	$Setup (anticommutativeprefix = psi)$

$[anticommutativeprefix = \{_λ, ψ\}]$

(22)

>	$am ≔ Annihilation (psi, notation = explicit)$

$am ≔ {a^{-}}_{ψ_{1}}$

(23)

>	$ap ≔ Creation (psi, notation = explicit)$

$ap ≔ {a^{+}}_{ψ_{1}}$

(24)

The related occupation number operator:

>	$N ≔ ap &period; am$

$N ≔ {a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}}$

(25)

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

>	$Ket (psi, 1)$

$|ψ_{1}⟩$

(26)

>	$am &period;$

$|ψ_{0}⟩$

(27)

>	$am &period;$

$0$

(28)

Increasing the occupation number,

>	$ap &period;$

$|ψ_{1}⟩$

(29)

>	$ap &period;$

$0$

(30)

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

>	${am}^{2}$

${a^{-}}_{ψ_{1}}^{2}$

(31)

>	$Simplify ({am}^{2})$

$0$

(32)

>	$Simplify ({ap}^{2})$

$0$

(33)

The occupation number operator is also idempotent:

>	$N^{} N N = N$

${a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}} {a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}} {a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}} = {a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}}$

(34)

These expressions can now be simplified:

>	$Simplify (%)$

${a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}} = {a^{+}}_{ψ_{1}} {a^{-}}_{ψ_{1}}$

(35)

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

>	$with (Vectors) &colon;$

>	$\vec{B0} \cdot (\vec{v} &x (\vec{B0} + \vec{B1}))$

$\vec{B0} \cdot (\vec{v} \times (\vec{B0} + \vec{B1}))$

(36)

>	$Simplify ()$

$\vec{B0} \cdot (\vec{v} \times \vec{B1})$

(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

>	$restart &semi; with (Physics) &colon; Setup (mathematicalnotation = true) &semi;$

$[mathematicalnotation = true]$

(38)

Define a contravariant 4-vector with components $[p__x, p__y, p__z, p__t] &colon;$

>	$Define (p [~ mu] = [p__x, p__y, p__z, p__t])$

$Defined objects with tensor properties$

$\{γ_{μ}, σ_{μ}, \partial_{μ}, g_{μ, ν}, p_{}^{μ}, δ_{μ, ν}, ε_{α, β, μ, ν}\}$

(39)

You can retrieve the components in different ways:

>	$p [~ mu] = Library :- TensorComponents (p [~ mu])$

$p_{}^{μ} = [p__x, p__y, p__z, p__t]$

(40)

>	$p [mu] = Library :- TensorComponents (p [mu])$

$p_{μ} = [- p__x, - p__y, - p__z, p__t]$

(41)

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

>	$p [~ 1] \neq p [1]$

$p__x \neq - p__x$

(42)

You can compute with $p_{}^{μ}$ algebraically; p[~mu] will return its components only when the index assumes integer values $0 \leq μ \leq 4.$

>	$p [mu] p [nu] ep_[mu, nu, alpha, beta]$

$ε_{α, β, μ, ν} p_{}^{μ} p_{}^{ν}$

(43)

>	$Simplify ()$

$0$

(44)

>	$p {[mu]}^{2}$

$p_{μ} p_{}^{μ}$

(45)

>	$SumOverRepeatedIndices ()$

${p__t}^{2} - {p__x}^{2} - {p__y}^{2} - {p__z}^{2}$

(46)

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

>	$Define (A, B, F, G)$

$Defined objects with tensor properties$

$\{A, B, F, G, γ_{μ}, σ_{μ}, \partial_{μ}, g_{μ, ν}, p_{}^{μ}, δ_{μ, ν}, ε_{α, β, μ, ν}\}$

(47)

A substitution equation:

>	$A_{μ} = G_{ν, α} A_{α} F_{μ, ν}$

$A_{μ} = G_{ν, α} A_{}^{α} F_{μ}^{ν}$

(48)

Substitute into $A_{ν} A_{}^{ν}$ : the free indices of (48) are taken as parameters, repeated indices in the substitution equation do not repeat more than once in the result:

>	$Library :- SubstituteTensor (, A_{ν} A_{}^{ν})$

$G_{β, α} A_{}^{α} F_{ν}^{β} G_{λ, κ} A_{}^{κ} F_{}^{ν, λ}$

(49)

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

>	$Library :- SubstituteTensor (A_{μ} (X) = B_{μ} (X), A_{ν} (Y))$

$B_{ν} (Y)$

(50)

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

>	$A [mu] B [nu] = G [mu, nu]$

$A_{μ} B_{ν} = G_{μ, ν}$

(51)

into,

>	$A [alpha] F [mu, nu] B [beta] A [rho] B [rho] G [mu, nu]$

$A_{α} F_{μ, ν} G_{}^{μ, ν} B_{β} A_{ρ} B_{}^{ρ}$

(52)

>	$Library :- SubstituteTensor (,)$

$G_{α, β} G_{ρ}^{ρ} F_{μ, ν} G_{}^{μ, ν}$

(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

>	$restart &semi; with (Physics) &colon; Setup (mathematicalnotation = true)$

$[mathematicalnotation = true]$

(56)

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

>	$(%Fundiff = Fundiff) (conjugate (f (x)), f (y))$

$(\frac{δ}{δ f (y)}) \overset{&conjugate0;}{f (x)} = 0$

(57)

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

>	$with (Vectors) &colon;$

>	$(%Fundiff = Fundiff) (%Gradient (f (x)), f (y))$

$(\frac{δ}{δ f (y)}) \nabla f (x) = δ' (x - y) \overset{\land}{i}$

(58)

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

>	$Coordinates (Q = [X, Y, Z, T])$

$Default differentiation variables for d_, D_ and dAlembertian are: \{Q = (X, Y, Z, T)\}$

$Systems of spacetime Coordinates are: \{Q = (X, Y, Z, T)\}$

$\{Q\}$

(59)

The Action for a system:

>	$S ≔ Intc (Norm {(%Gradient (Phi (x, y, z, t)))}^{2}, x, y, z, t)$

$S ≔ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {‖\nabla Φ (x, y, z, t)‖}^{2} &DifferentialD; x &DifferentialD; y &DifferentialD; z &DifferentialD; t$

(60)

The equations of motion through functional differentiation:

>	$(%Fundiff = Fundiff) (S, Phi (X, Y, Z, T))$

$(\frac{δ}{δ Φ (Q)}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {‖\nabla Φ (x, y, z, t)‖}^{2} &DifferentialD; x &DifferentialD; y &DifferentialD; z &DifferentialD; t = - \frac{{&DifferentialD;}^{2}}{&DifferentialD; Z^{2}} \overset{&conjugate0;}{Φ (Q)} - \frac{{&DifferentialD;}^{2}}{&DifferentialD; Y^{2}} \overset{&conjugate0;}{Φ (Q)} - \frac{{&DifferentialD;}^{2}}{&DifferentialD; X^{2}} \overset{&conjugate0;}{Φ (Q)}$

(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.

>	$S ≔ Intc (- \frac{1}{2} abs {(Gradient (Φ (x, y, z, t)))}^{2}, x, y, z, t)$

$S ≔ \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (- \frac{\frac{&DifferentialD;}{&DifferentialD; x} \overset{&conjugate0;}{Φ (x, y, z, t)} (\frac{\partial}{\partial x} Φ (x, y, z, t))}{2} - \frac{\frac{&DifferentialD;}{&DifferentialD; y} \overset{&conjugate0;}{Φ (x, y, z, t)} (\frac{\partial}{\partial y} Φ (x, y, z, t))}{2} - \frac{\frac{&DifferentialD;}{&DifferentialD; z} \overset{&conjugate0;}{Φ (x, y, z, t)} (\frac{\partial}{\partial z} Φ (x, y, z, t))}{2}) &DifferentialD; x &DifferentialD; y &DifferentialD; z &DifferentialD; t$

(62)

The corresponding field equation:

>	$Fundiff (S, Φ (X, Y, Z, T)) = 0$

$\frac{\frac{{&DifferentialD;}^{2}}{&DifferentialD; Z^{2}} \overset{&conjugate0;}{Φ (Q)}}{2} + \frac{\frac{{&DifferentialD;}^{2}}{&DifferentialD; Y^{2}} \overset{&conjugate0;}{Φ (Q)}}{2} + \frac{\frac{{&DifferentialD;}^{2}}{&DifferentialD; X^{2}} \overset{&conjugate0;}{Φ (Q)}}{2} = 0$

(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

>	$restart &semi; with (Physics) &colon; Setup (mathematicalnotation = true)$

$[mathematicalnotation = true]$

(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):

>	$g_[[28, 58.2, 1]]$

$Systems of spacetime Coordinates are: \{X = (r, ξ, ξ1, u)\}$

$Default differentiation variables for d_, D_ and dAlembertian are: \{X = (r, ξ, ξ1, u)\}$

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

$Parameters: [κ0, Q0, Q01, m0, b, Q]$

$Comments: All t&ExponentialE;nsor compon&ExponentialE;nts gⅈv&ExponentialE;n wⅈth r&ExponentialE;sp&ExponentialE;ct to th&ExponentialE; anholonomⅈc fram&ExponentialE;$

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

$g_[28]$

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

$[28, 46, 1] = [PrimaryDescription = EinsteinMaxwell, SecondaryDescription = [RobinsonTrautman], Comments = [Case 1 of 2]]$

$____________________________________________________________$

$[28, 46, 2] = [PrimaryDescription = EinsteinMaxwell, SecondaryDescription = [RobinsonTrautman, Stationary], Comments = [Case 2 of 2]]$

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

$[28, 73, 1] = [Authors = [Frolov and Khlebnikov (1975)], PrimaryDescription = PureRadiation, SecondaryDescription = [RobinsonTrautman], Comments = [The coordinates xi and xi1 are complex conjugates]]$

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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

$____________________________________________________________$

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 $\vec{A}, \vec{B}$ , are now implemented and expressed using the dot (scalar) product, as in ${[\vec{A}, \vec{B}]}_{-} = \vec{A} \cdot \vec{B} - \vec{B} \cdot \vec{A}$

•	If two noncommutative operators $a$ and $S$ satisfy ${[a^{†}, S]}_{-} = 0$ , then the commutator ${[a, S^{†}]}_{-}$ is automatically taken equal to 0; if in addition $S$ is Hermitian, then ${[a, S]}_{-}$ is also automatically taken equal to zero.

Examples

>	$restart &semi; with (Physics) &colon; Setup (mathematicalnotation = true)$

$[mathematicalnotation = true]$

(65)

Commutator and AntiCommutator now also operate on matrices:

>	$M__1 ≔ Matrix (2, 2, [a, b, c, d])$

$M__1 ≔ [\begin{array}{c} a & b \\ c & d \end{array}]$

(66)

>	$M__2 ≔ Matrix (2, 2, [alpha, beta, gamma, delta])$

$M__2 ≔ [\begin{array}{c} α & β \\ γ & δ \end{array}]$

(67)

>	$(%Commutator = Commutator) (M__1, M__2)$

${[[\begin{array}{c} a & b \\ c & d \end{array}], [\begin{array}{c} α & β \\ γ & δ \end{array}]]}_{-} = [\begin{array}{c} b γ - c β & a β - α b + b δ - β d \\ - γ a + c α - δ c + d γ & - b γ + c β \end{array}]$

(68)

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

>	$with (Vectors) &colon; Setup (op = \{A_, B_\})$

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

$[quantumoperators = \{\vec{A}, \vec{B}\}]$

(69)

>	$(Commutator = expand &commat; Commutator) (A_, B_)$

${[\vec{A}, \vec{B}]}_{-} = \vec{A} \cdot \vec{B} - \vec{B} \cdot \vec{A}$

(70)

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

>	$Setup (anticommutativeprefix = psi)$

$[anticommutativeprefix = \{_λ, ψ\}]$

(71)

>	$for j to 4 do ap [j] ≔ Creation (psi, j, notation = explicit) &semi; am [j] ≔ Annihilation (psi, j, notation = explicit) end do &semi;$

${ap}_{1} ≔ {a^{+}}_{ψ_{1}}$

${am}_{1} ≔ {a^{-}}_{ψ_{1}}$

${ap}_{2} ≔ {a^{+}}_{ψ_{2}}$

${am}_{2} ≔ {a^{-}}_{ψ_{2}}$

${ap}_{3} ≔ {a^{+}}_{ψ_{3}}$

${am}_{3} ≔ {a^{-}}_{ψ_{3}}$

${ap}_{4} ≔ {a^{+}}_{ψ_{4}}$

${am}_{4} ≔ {a^{-}}_{ψ_{4}}$

(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.

>	$Setup (algebra)$

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

$[algebrarules = \{{[{a^{-}}_{ψ_{1}}, {a^{+}}_{ψ_{1}}]}_{+} = 1, {[{a^{-}}_{ψ_{2}}, {a^{+}}_{ψ_{2}}]}_{+} = 1, {[{a^{-}}_{ψ_{3}}, {a^{+}}_{ψ_{3}}]}_{+} = 1, {[{a^{-}}_{ψ_{4}}, {a^{+}}_{ψ_{4}}]}_{+} = 1\}]$

(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.

>	$(%AntiCommutator = AntiCommutator) ({ap}_{1}, {am}_{1} {am}_{2} {ap}_{2})$

${[{a^{+}}_{ψ_{1}}, {a^{-}}_{ψ_{1}} {a^{-}}_{ψ_{2}} {a^{+}}_{ψ_{2}}]}_{+} = {a^{-}}_{ψ_{2}} {a^{+}}_{ψ_{2}}$

(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:

>	$P ≔ ap [2] am [2] am [1] ap [1]$

$P ≔ {a^{+}}_{ψ_{2}} {a^{-}}_{ψ_{2}} {a^{-}}_{ψ_{1}} {a^{+}}_{ψ_{1}}$

(75)

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

>	$ApAm ≔ [seq (ap [j], j = 1 .. 2), seq (am [j], j = 1 .. 2)]$

$ApAm ≔ [{a^{+}}_{ψ_{1}}, {a^{+}}_{ψ_{2}}, {a^{-}}_{ψ_{1}}, {a^{-}}_{ψ_{2}}]$

(76)

>	$Library :- SortProducts (P, ApAm, useanticommutator)$

${a^{+}}_{ψ_{1}} {a^{+}}_{ψ_{2}} {a^{-}}_{ψ_{1}} {a^{-}}_{ψ_{2}} + {a^{+}}_{ψ_{2}} {a^{-}}_{ψ_{2}}$

(77)

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