quantum mechanics and permutation invariants of finite groups

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Quantum mechanics and permutation invariants offinite groupsTo cite this article V V Kornyak 2013 J Phys Conf Ser 442 012050

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Quantum mechanics and permutation invariants of

finite groups

V V Kornyak

Laboratory of Information Technologies Joint Institute for Nuclear Research 141980 DubnaRussia

E-mail kornyakjinrru

Abstract We study quantum behavior from a constructive ldquofiniterdquo point of view since theintroduction of continuum or other actual infinities into physics poses serious conceptual andtechnical difficulties without any need for these concepts in physics as an empirical scienceTaking this approach we can show that the quantum-mechanical problems can be formulatedin the invariant subspaces of permutation representations of finite groups while the quantuminterferences occur as phenomena that are observable in these subspaces The scalar productsin the invariant subspaces (which are needed for formulating the Born rule mdash the mainpostulate of quantum mechanics that links mathematical description with experiment) arelinear combinations of independent bilinear invariant forms of the permutation representationA complete set of such forms for any permutation group can be easily calculated by asimple algorithm Slightly more sophisticated algorithms are required for expressing quantumobservables in terms of these forms

1 IntroductionQuantum behavior is a natural consequence of symmetries of physical systems In fact itis a manifestation of the fundamental impossibility to trace the identity of indistinguishableobjects in the process of their evolution The only that can be extracted from any physicalobservation is information about invariant relations and quantities which describe ensembles ofsuch objects Thus the fundamental problem is to study invariants of groups of symmetriesof physical systems Invariant bilinear forms ie scalar products are particularly importantfor quantum physics These forms in accordance with the Born rule provide the link betweenmathematical description and experimental observations

Physics as an empirical science is insensitive to whether finite or infinite concepts are usedin formulating its laws Moreover effective modeling is possible only when the problem isformulated in finite terms From the theory of quantum computing it is known that any unitaryoperator can be presented with arbitrary precision by a finite combination of matrix operatorsfrom a finite set of universal quantum gates Such universal sets of gates generate an infinitegroup Ginfin which is everywhere dense in the full unitary group By the theorem of Malrsquocev [1]the group Ginfin mdash as being a finitely generated matrix group mdash is residually finite [2] ie thereis a rich set of nontrivial homomorphisms from Ginfin to finite groups Combining these facts wecan replace the full unitary group by some finite group G in any particular problem withoutdistorting its physical content

DICE2012 IOP PublishingJournal of Physics Conference Series 442 (2013) 012050 doi1010881742-65964421012050

Content from this work may be used under the terms of the Creative Commons Attribution 30 licence Any further distributionof this work must maintain attribution to the author(s) and the title of the work journal citation and DOI

Published under licence by IOP Publishing Ltd 1

It is well known that any linear representation of a finite group is unitary1 Any unitaryrepresentation of a finite group is a subrepresentation of some permutation representation LetU be a representation of the group G in a K-dimensional Hilbert space HK Then U can beembedded into a permutation representation P of G in an N-dimensional Hilbert spaceHN whereN ge K The representation P is equivalent to an action of G on a set of things Ω = ω1 ωNby permutations In the proper case ie when N gt K the embedding has the structure

Tminus1PT =

1

V

HNminusK

UHK

HN = HNminusK oplusHK

where 1 is the trivial one-dimensional representation mandatory for any permutation represen-tation V is a complementary subrepresentation which may or may not be T is a matrixof transition from the basis of the permutation representation P to the basis in which thepermutation space HN is split into the invariant subspaces HNminusK and HK For brevity we willrefer to this basis as ldquoquantum basisrdquo The data in the spaces HK and HNminusK are independentsince both spaces are invariant subspaces of the space HN So we can consider the data in HNminusKas ldquohidden parametersrdquo with respect to the data in HK

A trivial approach would be to set arbitrary (eg zero) data in the complementary subspaceHNminusK This approach is not interesting since it is not falsifiable by means of the standardquantum mechanics In fact it leads to the standard quantum mechanics modulo the empiricallyunobservable distinction between the ldquofiniterdquo and the ldquoinfiniterdquo The only difference is technicalwe can replace the linear algebra in the K-dimensional space HK by permutations of N things

A more promising approach requires some changes in the concept of quantum amplitudes Weassume [3 4] that they are projections onto invariant subspaces of the vectors of multiplicities(ldquooccupation numbersrdquo) of elements of the set Ω on which the group G acts by permutationsThe vectors of multiplicities

|n〉 =

n1nN

are elements of the module HN = NN where N = 0 1 2 is the semiring of natural numbersWe start with the natural permutation representation of G in the module HN In order toturn the module HN into the Hilbert space HN it is sufficient just to add roots of unity mdashan algebraic incarnation of periodicity mdash to the natural numbers Linear combinations of Pthroots of unity with natural coefficients form a semiring NP The period P (it is called alsoconductor) depends on the structure of the group G Generally it is a multiple of some divisorof the exponent of G which is defined as the least common multiple of the orders of elements ofG We will always assume that P ge 2 In this case using the standard identities for the rootsof unity we can introduce the negative numbers and thus NP becomes a ring of cyclotomicintegers To complete the conversion of the module HN into the Hilbert space HN we introducethe cyclotomic field QP as a field of fractions of the ring NP If P ge 3 then QP is a densesubfield of the field of complex numbers C In particular this gives a trivial explanation of whythe complex numbers are so important in quantum mechanics In fact the algebraic propertiesof elements of QP are quite sufficient for all our purposes so we can forget the possibility toembed QP into C

The connection between mathematical description and observation is provided by the Bornrule the probability to register a particle described by the amplitude |ψ〉 by an apparatus tuned

1 An additional advantage of the transition to finite groups is a quite natural explanation of the ldquounitarityrdquo ofquantum mechanics it follows from this trivial property of finite groups


2

to the amplitude |φ〉 is

P(φ ψ) =|〈φ | ψ〉|2

〈φ | φ〉〈ψ | ψ〉

In the ldquofiniterdquo background the only reasonable interpretation of probability is the frequencyinterpretation the probability is the ratio of the number of ldquofavorablerdquo combinations to the totalnumber of combinations So we expect that P(φ ψ) must be a rational number if everything isarranged correctly

To calculate via the Born rule quantum interferences of projections of natural amplitudes itis sufficient to know the expressions for scalar products in invariant subspaces of the permutationrepresentation These expressions can be obtained by straightforward computation using thematrix of transition from the ldquopermutationrdquo to ldquoquantumrdquo basis Unfortunately there isno satisfactory algorithm for splitting a module over an associative algebra into irreduciblesubmodules mdash and hence for computing the corresponding transition matrix mdash over importantto us number systems of zero characteristic2 However we could try to cope with these difficultiesif we could get the expressions for the scalar products directly ie without the use of transitionmatrices We describe here one of the possible approaches This approach can be used forsplitting invariant forms of representations for which one can easily calculate the centralizerring Permutation representations belong just to this type of representations

2 Basis of permutation invariant formsFor a start we construct the set of all invariant bilinear forms in the permutation basis Supposethat the group G equiv G (Ω) that acts on the set Ω = ω1 ωN is generated by K elementsg1 gk gK Clearly it suffices to verify invariance of the form with respect to theseelements For convenience the set Ω is identified canonically with the set of indices of itselements Ω = ~N = 1 N Denote matrices of permutation representation of the generatorsby symbols Pk = P (gk) Obviously Pk

T = P(gminus1k

) The condition of invariance of the bilinear

form A =(aij)

under the group G can be written as a system of matrix equations

A = PkAPkT 1 le k le K

It is easy to check that in terms of components these equations have the form

aij = aigkjgk (1)

Thus the basis of all invariant bilinear forms is in one-to-one correspondence with the set oforbits ∆1 ∆R of the action of G on the Cartesian product Ω times Ω Orbits of the group onthe product Ωtimes Ω called orbitals play an important role in the theory of permutation groupsand their representations [6 7] If the group is transitive then there is a single orbital whichconsists of all pairs of the form (i i) Such an orbital is called trivial or diagonal The numberof orbitals R is called the rank of the group G Each orbital ∆r can be associated with

(i) a directed graph whose vertices are the elements of Ω and edges are pairs (i j) isin ∆r

(ii) an Ntimes N matrix A (∆r) which is constructed according to the rule

A (∆r)ij =

1 if (i j) isin ∆r


Properties of the graphs of orbitals reflect important properties of the groups For examplea transitive permutation group is primitive if and only if the graphs of all its non-trivial orbitalsare connected

2 The most practical and popular algorithm called MeatAxe [5] is designed only for algebras over finite fields


3

Matrices of orbitals form a basis of the centralizer ring of permutation representation of GThis ring plays an important role in the theory of group representations We will denote it byZR equiv ZR (G (Ω)) The multiplication table of basis elements of the ring ZR has the form

A (∆p)A (∆q) =

Rsumr=1

arpqA (∆r)

where all the structure constants arpq are natural numbersNote that the centralizer ring can be obtained via slightly different approach [8] which is

originated from the works of Schur The approach is based on the study of orbits of stabilizersof points in the set Ω Recall that the stabilizer of the element i isin Ω is the maximal subgroupGi of the group G that fixes this element ie g isin Gi hArr ig = i In the general case thestabilizer Gi of a transitive group G acts on the set Ω intransitively generating a set of orbitsΛ1 = i Λ2 ΛR There is a trivial one-to-one correspondence between the orbits of thestabilizer (sometimes they are called suborbits) and the orbitals (see eg [6 7]) Therefore thenumber R of orbits of the stabilizer Gi coincides with the rank of the permutation group G (Ω)Via the orbits of stabilizers the basis matrices of the centralizer ring are constructed as followsWe consider the orbits of the stabilizer G1 Each orbit Λ of the stabilizer G1 is associated withthe matrix A (Λ) by the rule

A (Λ)ij =

1 if there exist g isin G and l isin Λ such that 1g = i and lg = j

0 otherwise

Technical advantage of this approach is the use of the isomorphism between the orbits ofstabilizers of different points initially we construct the first row of the matrix using the orbits ofthe stabilizer of the point 1 and then the remaining rows are created via the group translations

We will call the matrices of orbitals Ar equiv A (∆r) basic forms since any permutation invariantbilinear form can be presented by their linear combination

Algorithm for computing the basic forms is reduced to construction of orbitals in accordancewith (1) In a few words it scans the elements of the set Ω times Ω in some say lexicographicorder and distributes these elements over the equivalence classes The output of the algorithmis a complete basis A1 AR of permutation invariant bilinear forms The algorithm is quitesimple Our implementation is just a few lines in C

The following identity follows directly from the construction

A1 +A2 + middot middot middot+AR = LTL = JN equiv

1 1 middot middot middot 11 1 middot middot middot 1

1 1 middot middot middot 1

where L is a covector of the form

N︷︸︸︷(1 1 1) and JN is an Ntimes N ldquomatrix of onesrdquo

Let us illustrate the algorithm with the example of the group Z3 acting on the set Ω = 1 2 3The group is generated by a single element for example g1 = (1 2 3) We need to distributethe set of pairs of indices

(1 1) (1 2) (1 3)(2 1) (2 2) (2 3)(3 1) (3 2) (3 3)

(2)


4

over the equivalence classes in accordance with (1) If we start from the top left corner of table(2) and will look for untreated pairs in the lexicographic order the orbitals will be constructedin the following order

∆1 = (1 1) (2 2) (3 3) ∆2 = (1 2) (2 3) (3 1) ∆3 = (1 3) (2 1) (3 2)

The corresponding complete basis of invariant forms reads

A1 =

1 middot middotmiddot 1 middotmiddot middot 1

A2 =

middot 1 middotmiddot middot 1

1 middot middot

A3 =

middot middot 1

1 middot middotmiddot 1 middot

Note that in general if we start the algorithm with the pair (1 1) and G is transitive then thefirst basis form will always be the identity matrix A1 = IN corresponding to the trivial orbital

3 Permutation invariant forms and decomposition into irreducible componentsDespite the ease of obtaining the permutation invariant forms they are sufficiently informativeConsider the decomposition of permutation representation into irreducible components usingthe transformation matrix T This decomposition for transitive groups has the form

Tminus1P(g)T =

1

Im2 otimesU2(g)

ImkotimesUk(g)

ImNIrrotimesUNIrr

(g)

g isin G

Here NIrr is the total number of different irreducible representations Uk (U1 equiv 1) of the groupG which are contained in the permutation representation P mk is the multiplicity of thesubrepresentations Uk in the representation P otimes denotes the Kronecker product of matrices Imis an mtimesm identity matrix

The most general permutation invariant form is a linear combination of the basic forms

A = a1A1 + a2A2 + middot middot middot+ aRAR (3)

where the coefficients ai are elements of some abelian number field3 F which is defined concretelyin the computations to be described below

It is easy to show (see [6 8]) that in a basis which splits the permutation representation intoirreducible components the form in (3) takes the form

Tminus1AT =

B1

B2 otimes Id2

Bk otimes Idk

BNIrrotimes IdNIrr

(4)

3 An abelian number field is an extension of the field Q with abelian Galois group Due to the KroneckerndashWebertheorem any such field is a subfield of some cyclotomic field QP


5

Here Bk is an mk times mk matrix whose elements are linear combinations of the coefficients aifrom (3) while mk is a multiplicity of the irreducible component Uk dk is the dimension ofUk The structure of matrix (4) implies that the rank R of the group (ie the dimension of thecentralizer ring) is equal to the sum of the squares of the multiplicities R = 1+m2

2 + middot middot middot+m2NIrr

Let us now consider the determinant det (A) In terms of the variables a1 aR this

determinant is a homogeneous polynomial of degree N Since the determinant of a form doesnot depend on the choice of basis and the determinant of a block diagonal matrix is the productof determinants of its blocks from decomposition (4) it follows

det (A) = det (B1) det (B2)d2 middot middot middot det (Bk)dk middot middot middot det (BNIrr)dNIrr

Here we have used the identity det (X otimes Y ) = det (X)m det (Y )n for the Kronecker product ofan ntimes n matrix X and an mtimesm matrix Y

It is clear that det (Bk) is a homogeneous polynomial of degree mk in the variables a1 aRdet (Bk) = Ek (a1 aR) Thus we have the following

Proposition The determinant of linear combination (3) has the following decompositioninto factors over a certain ring of cyclotomic integers

detRsumi=1

aiAi =

NIrrprodk=1

Ek (a1 aR)dk degEk (a1 aR) = mk (5)

where Ek is an irreducible polynomial corresponding to the irreducible component Uk in therepresentation P Recall that NIrr is the number of different irreducible components in P dk isdimension of Uk mk is the multiplicity of Uk in P

From this proposition the idea of an algorithm to compute the invariant forms in irreduciblesubspaces of the permutation representation follows

We must first calculate the polynomial det (A) This is a relatively simple task In particularalgorithms based on the Gaussian elimination have a cubic complexity in the size of the matrix

Then the polynomial det(A) must be decomposed into a maximum number of irreduciblefactors Note that advanced algorithms of polynomial factorization automatically determinean algebraic extension of rationals that guarantees maximal factorization Such algorithmsare called algorithms of ldquoabsolute factorizationrdquo There are many practical implementations ofpolynomial factorization algorithms with different estimates of the complexity4

Thus constructing decomposition (5) is an algorithmically realizable task Solving it weobtain the complete information about the dimensions and multiplicities of all irreduciblesubrepresentations

The next natural step is to try to compute explicitly invariant forms Bk in the irreduciblesubspaces of the permutation representation To do this we exclude from consideration thefactor Ek (a1 aR) related to the component Bk and equate to zero the other factors Thatis we write the system of equations

E1 = middot middot middot = Ek = middot middot middot = ENIrr= 0 (6)

31 Example with irreducible components of multiplicity oneIf all the multiplicities mi = 1 (under this condition R = NIrr and the centralizer ring ZR iscommutative) then all the polynomials Ei are linear In this case the computation of the scalarproducts in the invariant subspaces can easily be completed

4 To improve efficiency we can try to write a specialized algorithm that takes into account the fact that in ourcase the polynomials are factored over the rings of cyclotomic integers whose conductors are divisors of groupexponents However the benefits of such work require a separate study


6

As an example consider the group SL(2 3) defined as a group of special linear transformationsof two-dimensional space over the field of three elements F3 This group is used in particle physicswhere it is often denoted by Tprime or 2T since it is a double cover of the symmetry group T sim= A4

of a tetrahedron We will consider its faithful permutation action of degree 8 which can begenerated for example by the following two permutations

g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)

The 8-dimensional permutation representation will be denoted by 8The following four matrices mdash obtained via constructing orbitals mdash form a basis of the ring

ZR equiv ZR(SL(2 3)

(~8))

of permutation invariant forms

A1 =

1 middot middot middot middot middot middot middotmiddot 1 middot middot middot middot middot middotmiddot middot 1 middot middot middot middot middotmiddot middot middot 1 middot middot middot middotmiddot middot middot middot 1 middot middot middotmiddot middot middot middot middot 1 middot middotmiddot middot middot middot middot middot 1 middotmiddot middot middot middot middot middot middot 1

A2 =

middot 1 middot middot middot middot middot middot1 middot middot middot middot middot middot middotmiddot middot middot 1 middot middot middot middotmiddot middot 1 middot middot middot middot middotmiddot middot middot middot middot 1 middot middotmiddot middot middot middot 1 middot middot middotmiddot middot middot middot middot middot middot 1middot middot middot middot middot middot 1 middot

A3 =

middot middot 1 middot 1 middot 1 middotmiddot middot middot 1 middot 1 middot 1middot 1 middot middot middot 1 1 middot1 middot middot middot 1 middot middot 1middot 1 1 middot middot middot middot 11 middot middot 1 middot middot 1 middotmiddot 1 middot 1 1 middot middot middot1 middot 1 middot middot 1 middot middot

A4 =

middot middot middot 1 middot 1 middot 1middot middot 1 middot 1 middot 1 middot1 middot middot middot 1 middot middot 1middot 1 middot middot middot 1 1 middot1 middot middot 1 middot middot 1 middotmiddot 1 1 middot middot middot middot 11 middot 1 middot middot 1 middot middotmiddot 1 middot 1 1 middot middot middot

The determinant of their linear combination A = a1A1 + a2A2 + a3A3 + a4A4 is decomposedinto linear factors over the ring of cyclotomic integers N3

detA = (a1 + a2 + 3a3 + 3a4)

a1 minus a2 + (1 + 2r) a3 minus (1 + 2r) a42

a1 minus a2 minus (1 + 2r) a3 + (1 + 2r) a42

(a1 + a2 minus a3 minus a4)3

where r is a third primitive root of unity From this formula the structure of decomposition ofthe representation 8 into irreducible components is seen immediately

8 = 1oplus 2oplus 2prime oplus 3

Excluding sequentially the linear factors corresponding to the representations k = 122prime3and equating to zero the other factors we obtain four systems of three linear equations

Consider for example the subrepresentation 2 For this component the system of equations(6) takes the form

a1 + a2 + 3a3 + 3a4 = 0 (7)

a1 minus a2 minus (1 + 2r) a3 + (1 + 2r) a4 = 0 (8)

a1 + a2 minus a3 minus a4 = 0 (9)


7

The linear systems of this type consist of Rminus 1 equations but contain R variables Since thebilinear form (3) describes a non-degenerate scalar product the coefficient a1 at the diagonalbasis form can not vanish Thus one can treat a1 as a parameter and solve the system ofequations for the remaining variables In principle the coefficient a1 can be taken to be anarbitrary (nonzero) parameter since Bornrsquos probability is independent of its value However itis reasonable to choose the value

a1 = dkN (10)

for each irreducible component Under such a normalization the sum of scalar products ininvariant subspaces will be equal to the standard scalar product in the space of permutationrepresentation

Solving linear system (7mdash9) and setting a1 = 28 = 14 we obtain

a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12

Applying the same procedure to all other irreducible components we come to the followingset of forms defining scalar products in the invariant subspaces

B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)

It is easy to check that normalization (10) ensures the identity B1 + B2 + B2prime + B3 = A1 equiv I8Recall that in all such tasks B1 does not require calculation since the inner product in the

subspace of the trivial representation always has the form1

NJN

Let |n〉 = (n1 middot middot middot n8)T and |m〉 = (m1 middot middot middot m8)T be natural vectors in the permutationbasis and |Ψk〉 and |Φk〉 their projections onto the invariant subspaces where k = 122prime3The scalar products of these projections in the invariant subspaces can be expressed in terms ofthe natural vectors as

〈Φk |Ψk〉 = 〈m |Bk|n〉

For example for the subrepresentation 1 we have

〈Φ1 |Ψ1〉 =1

8(m1 +m2 + middot middot middot+m8) (n1 + n2 + middot middot middot+ n8)

It is clear that this expression never vanishes for natural vectors |n〉 and |m〉 In fact inthe general case the trivial one-dimensional subrepresentation contained in any permutationrepresentation can be interpreted as the ldquocounter of particlesrdquo since the permutation invariantsumN

i=1 ni corresponding to this subrepresentation is the total number of elements of the set Ω inthe ensemble

As for the other subrepresentations we can observe non-trivial ldquoquantum behaviorrdquo in theirinvariant subspaces In particular the equations 〈m |Bk|n〉 = 0 mdash the conditions for destructiveinterference mdash have infinitely many solutions in natural vectors for k = 22prime3

Note also that Bornrsquos probabilities computed via the forms B2 and B2prime take irrational valuesfor natural vectors This contradicts the idea that any probability in the finite background must


8

be rational The contradiction is resolved by the fact that the permutation action that weconsider is imprimitive ie it moves some nontrivial subsets of Ω as single units Such subsetsare called blocks In our example the block system (called also the system of imprimitivity)is the following [1 2] [3 4] [5 6] [7 8] Thus we can not take the subrepresentations 2 and2prime separately and must consider their sum 2 oplus 2prime instead The invariant form for this sumB2oplus2prime = 1

2 (A1 minusA2) does not contain irrationalities

32 The case of multiple subrepresentationsIn the case of multiple subrepresentations the situation becomes more complicated because of thenonlinearity of relations The question of whether it is possible to develop a general algorithmfor the case of multiple subrepresentations requires a deeper additional study In many concreteexamples using analogy to the case of single subrepresentations and some ad hoc tricks it ispossible to carry out the complete decomposition of scalar products into irreducible components

The source of nonlinearity is an excess in the number of parameters the definite values ofwhich are inessential for computing the Born probabilities As one can see from the structure ofdecomposition (4) any block of multiple components BkotimesIdk

contains mktimesmk such parametersOur convention (10) allows to fix mk diagonal elements of the matrix Bk So we need to fixsomehow the remaining m2

k minusmk parameters In some cases the examination of the structureof the centralizer ring helps to do this

As an illustrative example let us consider the Coxeter group A2 which is also the Weylgroup of the Lie group say SU (3) We will consider the natural action of A2 on its root systemΩ = 1 2 middot middot middot 6 The vectors of this root system form a two-dimensional regular hexagon Thegenerators of the action are for example g1 = (1 4)(2 3)(5 6) and g2 = (1 3)(2 5)(4 6)

Computation of orbitals gives the following basis of the centralizer ring

A1 =

1 middot middot middot middot middotmiddot 1 middot middot middot middotmiddot middot 1 middot middot middotmiddot middot middot 1 middot middotmiddot middot middot middot 1 middotmiddot middot middot middot middot 1

A2 =

middot 1 middot middot middot middotmiddot middot middot middot middot 1middot middot middot middot 1 middotmiddot middot 1 middot middot middotmiddot middot middot 1 middot middot1 middot middot middot middot middot

A3 =

middot middot 1 middot middot middotmiddot middot middot 1 middot middot1 middot middot middot middot middotmiddot 1 middot middot middot middotmiddot middot middot middot middot 1middot middot middot middot 1 middot

A4 =

middot middot middot 1 middot middotmiddot middot middot middot 1 middotmiddot middot middot middot middot 11 middot middot middot middot middotmiddot 1 middot middot middot middotmiddot middot 1 middot middot middot

A5 =

middot middot middot middot 1 middotmiddot middot 1 middot middot middotmiddot 1 middot middot middot middotmiddot middot middot middot middot 11 middot middot middot middot middotmiddot middot middot 1 middot middot

A6 =

middot middot middot middot middot 11 middot middot middot middot middotmiddot middot middot 1 middot middotmiddot middot middot middot 1 middotmiddot middot 1 middot middot middotmiddot 1 middot middot middot middot

Applying the algorithm of absolute factorization to the generic linear combination

A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6

we obtain the following decomposition

detA = (a1 + a2 minus a3 minus a4 minus a5 + a6)

(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2

3 minus a24 minus a2

5 + a26 minus a1a2 minus a1a6 minus a2a6 + a3a4 + a3a5 + a4a5

2

The structure of the action of the Weyl group A2 on its roots in terms of the permutationrepresentation follows immediately from (11) 6 = 1oplus 1primeoplus 2oplus 2 Here we have four irreducible


9

components and a six-dimensional centralizer ring Thus we need to eliminate the two extradegrees of freedom Basis matrices derived from the orbitals are (01)-matrices with disjoint setsof unit entries A result of addition of such matrices belongs to the same type of matrices Sothe natural idea is to sum up some of the matrices Ai in order to reduce their total numberRecall that if the centralizer ring is commutative then all the multiplicities are equal to oneSo we retain the commutative subset of matrices unchanged In our case such a subset consistsof the matrices A1 A2 and A6 ldquoKilling noncommutativityrdquo by replacing the remaining non-commuting matrices by their sum AΣ = A3 +A4 +A5 we write the linear combination of fourmatrices

Aprime = a1A1 + a2A2 + a6A6 + aΣAΣ

Now the determinant is factorizable to the linear factors over the cyclotomic integer ring N3

detAprime = (a1 + a2 + a6 minus 3aΣ)

(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62

where r is a third root of unity After the same manipulations as above we come to the followingset of scalar product forms in the invariant subspaces

B1 =1

6(A1 +A2 +A6 +AΣ) equiv 1

6J6

B1prime =1

6(A1 +A2 +A6 minusAΣ)

B2 =1

3A1 minus (1 + r)A2 + rA6

B2 =1

3A1 + rA2 minus (1 + r)A6

Here B2 and B2 are different coordinate presentations of the same form associated with theirreducible representation 2

4 ConclusionsElimination of actual infinities from the description of physical reality removes many technicaldifficulties This allows to focus on the content aspects of physical problems We have consideredsome implications of the idea that quantum mechanics as any reasonable physical theory mustallow effective modeling by finite means

Using mathematical arguments of general nature we can show that any quantum problemcan be reduced to permutations If we assume also that the entities which are subject to thepermutations have a physical meaning we come to a very simple and self-consistent pictureof the quantum behavior To study the consequences of this assumption we need to know theinner products in the invariant subspaces of permutation representations

In this paper we have considered a possible algorithmic approach to calculating theseinner products With this approach we can for any permutation representation obtain thefull information about the structure of its decomposition into irreducible components Ifall irreducible components have unit multiplicities then the invariant inner products can beeasily computed There are many observations which indicate that in the case of multiplesubrepresentations a reasonable algorithm is also possible


10

AcknowledgmentsThe work was partially supported by the grants 10-01-00200 from the Russian Foundation forBasic Research and 380220122 from the Ministry of Education and Science of the RussianFederation

References[1] Malrsquocev A 1940 On isomorphic matrix representations of infinite groups Mat Sbor 8(50)(3) 405ndash422

(Russian)[2] Magnus W 1969 Residually finite groups Bull Amer Math Soc 75(2) 305ndash316[3] Kornyak V V 2012 Permutation interpretation of quantum mechanics J Phys Conf Ser 343 012059

doi1010881742-65963431012059[4] Kornyak V V 2013 Classical and quantum discrete dynamical systems Phys Part Nucl 44(1) 47-91 (Preprint

arXiv12085734 [quant-ph])[5] Holt D F Eick B and OrsquoBrien E A 2005 Handbook of Computational Group Theory (Boca-RatonmdashLondonmdash

New-YorkmdashWashington-DC Chapman amp HallCRC)[6] Cameron P J 1999 Permutation Groups (Cambridge Cambridge University Press)[7] Dixon J D and Mortimer B 1996 Permutation Groups (Berlin Springer)[8] Wielandt H 1964 Finite Permutation Groups (New-YorkmdashLondon Academic Press)


11

Quantum mechanics and permutation invariants of

finite groups

V V Kornyak

Laboratory of Information Technologies Joint Institute for Nuclear Research 141980 DubnaRussia

E-mail kornyakjinrru

Abstract We study quantum behavior from a constructive ldquofiniterdquo point of view since theintroduction of continuum or other actual infinities into physics poses serious conceptual andtechnical difficulties without any need for these concepts in physics as an empirical scienceTaking this approach we can show that the quantum-mechanical problems can be formulatedin the invariant subspaces of permutation representations of finite groups while the quantuminterferences occur as phenomena that are observable in these subspaces The scalar productsin the invariant subspaces (which are needed for formulating the Born rule mdash the mainpostulate of quantum mechanics that links mathematical description with experiment) arelinear combinations of independent bilinear invariant forms of the permutation representationA complete set of such forms for any permutation group can be easily calculated by asimple algorithm Slightly more sophisticated algorithms are required for expressing quantumobservables in terms of these forms

1 IntroductionQuantum behavior is a natural consequence of symmetries of physical systems In fact itis a manifestation of the fundamental impossibility to trace the identity of indistinguishableobjects in the process of their evolution The only that can be extracted from any physicalobservation is information about invariant relations and quantities which describe ensembles ofsuch objects Thus the fundamental problem is to study invariants of groups of symmetriesof physical systems Invariant bilinear forms ie scalar products are particularly importantfor quantum physics These forms in accordance with the Born rule provide the link betweenmathematical description and experimental observations

Physics as an empirical science is insensitive to whether finite or infinite concepts are usedin formulating its laws Moreover effective modeling is possible only when the problem isformulated in finite terms From the theory of quantum computing it is known that any unitaryoperator can be presented with arbitrary precision by a finite combination of matrix operatorsfrom a finite set of universal quantum gates Such universal sets of gates generate an infinitegroup Ginfin which is everywhere dense in the full unitary group By the theorem of Malrsquocev [1]the group Ginfin mdash as being a finitely generated matrix group mdash is residually finite [2] ie thereis a rich set of nontrivial homomorphisms from Ginfin to finite groups Combining these facts wecan replace the full unitary group by some finite group G in any particular problem withoutdistorting its physical content


Content from this work may be used under the terms of the Creative Commons Attribution 30 licence Any further distributionof this work must maintain attribution to the author(s) and the title of the work journal citation and DOI

Published under licence by IOP Publishing Ltd 1


Tminus1PT =

1

V

HNminusK

UHK





|n〉 =

n1nN





2


P(φ ψ) =|〈φ | ψ〉|2

〈φ | φ〉〈ψ | ψ〉




T = P(gminus1k


form A =(aij)




aij = aigkjgk (1)




A (∆r)ij =






3


A (∆p)A (∆q) =

Rsumr=1

arpqA (∆r)



A (Λ)ij =


0 otherwise











(1 1) (1 2) (1 3)(2 1) (2 2) (2 3)(3 1) (3 2) (3 3)

(2)


4


∆1 = (1 1) (2 2) (3 3) ∆2 = (1 2) (2 3) (3 1) ∆3 = (1 3) (2 1) (3 2)


A1 =


A2 =


1 middot middot

A3 =

middot middot 1




Tminus1P(g)T =

1

Im2 otimesU2(g)

ImkotimesUk(g)

ImNIrrotimesUNIrr

(g)

g isin G






Tminus1AT =

B1

B2 otimes Id2

Bk otimes Idk

BNIrrotimes IdNIrr

(4)



5









detRsumi=1

aiAi =

NIrrprodk=1












6



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11


Tminus1PT =

1

V

HNminusK

UHK





|n〉 =

n1nN





2


P(φ ψ) =|〈φ | ψ〉|2

〈φ | φ〉〈ψ | ψ〉




T = P(gminus1k


form A =(aij)




aij = aigkjgk (1)




A (∆r)ij =






3


A (∆p)A (∆q) =

Rsumr=1

arpqA (∆r)



A (Λ)ij =


0 otherwise











(1 1) (1 2) (1 3)(2 1) (2 2) (2 3)(3 1) (3 2) (3 3)

(2)


4


∆1 = (1 1) (2 2) (3 3) ∆2 = (1 2) (2 3) (3 1) ∆3 = (1 3) (2 1) (3 2)


A1 =


A2 =


1 middot middot

A3 =

middot middot 1




Tminus1P(g)T =

1

Im2 otimesU2(g)

ImkotimesUk(g)

ImNIrrotimesUNIrr

(g)

g isin G






Tminus1AT =

B1

B2 otimes Id2

Bk otimes Idk

BNIrrotimes IdNIrr

(4)



5









detRsumi=1

aiAi =

NIrrprodk=1












6



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11


P(φ ψ) =|〈φ | ψ〉|2

〈φ | φ〉〈ψ | ψ〉




T = P(gminus1k


form A =(aij)




aij = aigkjgk (1)




A (∆r)ij =






3


A (∆p)A (∆q) =

Rsumr=1

arpqA (∆r)



A (Λ)ij =


0 otherwise











(1 1) (1 2) (1 3)(2 1) (2 2) (2 3)(3 1) (3 2) (3 3)

(2)


4


∆1 = (1 1) (2 2) (3 3) ∆2 = (1 2) (2 3) (3 1) ∆3 = (1 3) (2 1) (3 2)


A1 =


A2 =


1 middot middot

A3 =

middot middot 1




Tminus1P(g)T =

1

Im2 otimesU2(g)

ImkotimesUk(g)

ImNIrrotimesUNIrr

(g)

g isin G






Tminus1AT =

B1

B2 otimes Id2

Bk otimes Idk

BNIrrotimes IdNIrr

(4)



5









detRsumi=1

aiAi =

NIrrprodk=1












6



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11


A (∆p)A (∆q) =

Rsumr=1

arpqA (∆r)



A (Λ)ij =


0 otherwise











(1 1) (1 2) (1 3)(2 1) (2 2) (2 3)(3 1) (3 2) (3 3)

(2)


4


∆1 = (1 1) (2 2) (3 3) ∆2 = (1 2) (2 3) (3 1) ∆3 = (1 3) (2 1) (3 2)


A1 =


A2 =


1 middot middot

A3 =

middot middot 1




Tminus1P(g)T =

1

Im2 otimesU2(g)

ImkotimesUk(g)

ImNIrrotimesUNIrr

(g)

g isin G






Tminus1AT =

B1

B2 otimes Id2

Bk otimes Idk

BNIrrotimes IdNIrr

(4)



5









detRsumi=1

aiAi =

NIrrprodk=1












6



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11


∆1 = (1 1) (2 2) (3 3) ∆2 = (1 2) (2 3) (3 1) ∆3 = (1 3) (2 1) (3 2)


A1 =


A2 =


1 middot middot

A3 =

middot middot 1




Tminus1P(g)T =

1

Im2 otimesU2(g)

ImkotimesUk(g)

ImNIrrotimesUNIrr

(g)

g isin G






Tminus1AT =

B1

B2 otimes Id2

Bk otimes Idk

BNIrrotimes IdNIrr

(4)



5









detRsumi=1

aiAi =

NIrrprodk=1












6



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11









detRsumi=1

aiAi =

NIrrprodk=1












6



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11



g1 = (1 5 3 2 6 4) (7 8) and g2 = (1 3 7 2 4 8) (5 6)


ZR equiv ZR(SL(2 3)

(~8))


A1 =


A2 =


A3 =


A4 =



detA = (a1 + a2 + 3a3 + 3a4)








a1 + a2 + 3a3 + 3a4 = 0 (7)




7


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11


a1 = dkN (10)



a2 = minus1

4 a3 = minus1 + 2r

12 a4 =

1 + 2r

12


B1 =1

8(A1 +A2 +A3 +A4) equiv 1

8J8

B2 =1

4

(A1 minusA2 minus

1 + 2r

3A3 +

1 + 2r

3A4

)

B2prime =1

4

(A1 minusA2 +

1 + 2r

3A3 minus

1 + 2r

3A4

)

B3 =3

8

(A1 +A2 minus

1

3A3 minus

1

3A4

)



NJN




〈Φ1 |Ψ1〉 =1







8









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11









A1 =


A2 =


A3 =


A4 =


A5 =


A6 =



A = a1A1 + a2A2 + a3A3 + a4A4 + a5A5 + a6A6



(a1 + a2 + a3 + a4 + a5 + a6) (11)a2

1 + a22 minus a2



2



9





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11





(a1 + a2 + a6 + 3aΣ)

a1 + ra2 minus (1 + r) a62

a1 minus (1 + r) a2 + ra62


B1 =1


6J6

B1prime =1


B2 =1


B2 =1







10








11








11

quantum mechanics and permutation invariants of finite groups

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