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Numerical Methods and Programming, 2009, Vol. 10 (http://num-meth.srcc.msu.ru)

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UDC 512.54; 519.1; 530.145

SYMMETRIES, GAUGE INVARIANCE AND QUANTIZATION IN DISCRETE MODELS1 V. V. Kornyak2

Di erent aspects of discrete symmetry analysis in application to deterministic and non-deterministic lattice models are considered. One of the main tools for our study are programs written in C. In the case of deterministic dynamical systems, such as cellular automata, the non-trivial connections between the lattice symmetries and dynamics are discussed. In particular, we show that the formation of moving soliton-like structures | analogs of \spaceships" in cellular automata or \generalized coherent states" in quantum physics | results from the existence of a non-trivial symmetry group. In the case of mesoscopic lattice models, we apply some algorithms exploiting the symmetries of the models to compute microcanonical partition functions and to search phase transitions. We also consider the gauge invariance in discrete dynamical systems and its connection with quantization. We propose a constructive approach to introduce quantum structures in discrete systems based on nite gauge groups. In this approach, quantization can be interpreted as the introduction of a gauge connection of a special kind. We illustrate our approach to quantization by a simple model and propose its generalization.

symmetric discrete formations (see, e.g., Fig. 1).

Keywords: symmetries of discrete systems, gauge principle, quantization. 1. Introduction. Discrete systems are widespread in applications. In particular, many nanostructures are

Fig. 1. Symmetries of 3-valent (hydro)carbon nanostructures: a) Tetrahedrane C4H4 (G = Sym(4), Gloc = D6 = Sym(3); b) Cubane C8 H8 (G = Z Sym(4), Gloc = D6 ; 2 c) Dodecahedrane C20H20 (G = Z Alt(5), Gloc = D6; d) Fullerene C60 (G = Z Alt(5), Gloc = Z ; 2 2 2 e) Toric graphene n m (G = Dn D2n, Gloc = Z ; n; m ! 1, G = (Z Z o D6 , Gloc = D6 ) 2 From a fundamental point of view, there are many philosophical and physical arguments that discreteness is more suitable for describing physics at small distances than the continuity that arises only as an approximation or as a logical limit in considering large collections of discrete structures. In 1912 Henri Poincare wrote 1]: \We now wonder not only whether the di erential equations of dynamics must be modi ed, but whether the laws of motion can still be expressed by means of di erential equations : : : . It is being asked whether it is not necessary to introduce discontinuities into the natural laws, not apparent ones but essential ones". Since that time, the number and weights of arguments supporting similar views have increased considerably. In this paper we consider deterministic and non-deterministic dynamical systems with non-trivial symmetries de ned on discrete spaces and evolving in discrete time. As a tool for our study, we are developing programs in C based on computer algebra and computational group theory methods 2]. 1 This work was supp orted by the RFBR grant 07{01{00660 and the grant 1027.2008.2 from the Ministry of Education and Science of the Russian Federation. 2 Lab oratory of Information Technologies, Joint Institute for Nuclear Research, 141980, Dubna, Russia; e-mail: kornyak@jinr.ru c Science Research Computing Center, Moscow State University


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form a 1) 2) 3) 4) as the

2. Basic constituents of discrete models. In this paper we consider the following constructions that

basis for all types of dynamical systems: a space X is a k-valent graph with the symmetry group G = Aut(X ) | space symmetries; the vertices x of X take values in a set with the symmetry group 6 Sym( ) | internal symmetries; the states of the whole system are functions (x) 2 X ; we de ne the whole system symmetry groups W | unifying the space G and the internal symmetries | equivalence classes of split group extensions 3] of the form 1 ! X ! W ! G ! 1;
F

(1)

5) action of the group W splits the set of states X into orbits of di erent sizes: X = Oi; i 6) evolution proceeds in discrete time t 2 Z= f: : : ; 1; 0; 1; : : : g ; 7) dynamics is determined by some evolution rule connecting the current state t(x) of the system with its prehistory t 1(x), t 2(x); t 3(x); : : : : For models with locally de ned evolution rules (such as, e.g., cellular automata or Ising model), the group of local symmetries Gloc is essential and is de ned as a stabilizer of a vertex x in the space group G: g 2 Gloc = StabG (x) means xg = x: Local rules are de ned on orbits of Gloc on edges from the neighborhoods of the points x: Figure 1 shows the symmetry groups G and Gloc 6 G for some carbon and hydrocarbon molecules. Let us give an explicit description of the whole symmetry group W: The equivalence classes in (1) are determined by arbitrary antihomomorphisms : G ! G. The term \antihomomorphism" means that (a) (b) = (ba): The equivalence is described by arbitrary function : G ! G: The explicit formulas for the main group operations | action 3 on X , multiplication, and inversion | are (x) (x); a = x (a) x (a) ; (x); a (x); b = (x (ab) 1 (b) (a) (x); a
1

x (ab) (a)
1

1 1

(b) ; ab ; :

=

x (a 1)

1

(a)

1

;a

Here (x); a 2 W with (x) 2 X and a 2 G, etc. Note that the standard direct and wreath products are obtained from this general construction by choosing (a) = 1 and (a) = a 1; respectively. As to the arbitrary function ; the choices (a) = 1 and (a) = a 1; respectively, are generally used in the literature. Thus, the action and multiplication take the following forms for the direct product X G: (x) (x); a = (x) (x) , (x); a (x); b = (x) (x); ab and for the wreath product oX G: (x) (x); a = (xa 1 ) (xa 1) , (x); a (x); b = (x) (xa); ab . 3. Deterministic and non-deterministic dynamics. The evolution rule of a deterministic (or causal ) dynamical system is a functional relation. This means that the current state is a function of the prehistory: (x) = F t 1(x); t 2(x); t 3(x); : : : ) : (2) Cel lular automaton is a typical example of a deterministic dynamical system. The causality imposes severe restrictions on the system dynamics 4]. In particular, for the rst-order4 functional relations: | dynamical trajectories pass group orbits in a non-decreasing order of orbit sizes, | periodic tra jectories lie within orbits of the same size. A mesoscopic lattice model is a type of non-deterministic dynamical systems. This is a special case of a Markov chain. In a mesoscopic lattice model, the transition from one state to any other is possible with a probability controlled by a Hamiltonian. A quantum system is another important type of non-deterministic dynamical systems. The probabilities of transitions between states are expressed in terms of complex-valued transition amplitudes. 4. Soliton-like structures in deterministic dynamics. For deterministic dynamics, the recurrence of dynamical tra jectory to the same group orbit is typical. Moreover, if the symmetry group W splits the state set X into nite number of orbits (this is the case for all systems we consider here), then after a lapse of time 3 We write group actions on the right. This, more intuitive, convention is adopted in b oth GAP and MAGMA | the most widespread computer algebra systems with advanced facilities for computational group theory. 4 This means that (2) takes the form t(x) = F t 1(x) :
t


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any tra jectory comes inevitably to a cycle over some nite sequence of orbits. This just means the formation of soliton-like structures. Namely, let us consider the evolution
t
0

(x) ! t1 (x) = At1 t

0

t

0

(x) :

(3)

X , then If the states at the moments t0 and t1 belong to the same orbit : t0 (x) 2 Oi and t0 (x) 2 Oi ; Oi evolution (3) can be replaced by the group action t1 (x) = t0 (x)w, w 2 W , i.e., the initial state t0 (x) is reproduced after some \movement " in the space X : Several examples (including continuous cases) of cycles over group orbits: | running waves (x vt) in mathematical physics | Galilei group; | \generalized coherent states " in quantum physics | unitary representations of Lie groups; | \spaceships " in cellular automata | lattice symmetries.

Fig. 2. Example of soliton-like structure Figure 2 illustrates the formation of \glider" | one of the \spaceships" in Conway's cellular automaton \Game of Life". \Glider" in Conway's \Life"is a cycle in two orbits of the square lattice symmetry group (the semidirect product of 2D translations and dihedral group D8): the con gurations '3 and '4 are obtained from '1 and '2 , respectively, by the same combination of downward shift, 90 clockwise rotation and re ection with respect to the vertical.

Fig. 3. Ising model on dodecahedron. Microcanonical distribution and \convex intruders" indicating mesoscopic phase transitions 5. Mesoscopic lattice models. The discrete symmetry analysis simpli es the manipulations with microcanonical ensembles and the search of phase transitions. This allows one to reveal subtle details in the behavior of mesoscopic models : in addition to distinct \convex intruder " denoted by A (a criterion of phase transition adopted in mesoscopy ), in Fig. 3 we see that the computation detects a subtle \intruder" B . 6. Discrete gauge principle. In fact, the gauge principle expresses the general idea that any observable data can be presented in di erent \frames" at di erent points and there should be some way to compare these data. At the set-theoretic level suitable for both the discrete and the continuous cases, the main concepts of the gauge principle can be reduced to the following elements: | a set X , space or space-time; | a set , local states; | the set X of -valued functions on X , the set of states of dynamical systems ;


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| a group W 6 Sym( X ) acting transitively on X , symmetries of the system ; | identi cation of data describing a dynamical system makes sense only modulo symmetries from W ; | having no a priori connection between data from X at di erent points x and y in time and space, we impose this connection (or paral lel transport 5]) explicitly as W -valued functions on edges of an abstract graph: P (x; y) 2 W , & (y) = (x)P (x; y); the connection P (x; y) has the obvious property P (y; x) = P (x; y) 1; e | a connection P (x; y) is called trivial if it can be expressed in terms of a function on the vertices of a e(x; y) = r(x) 1r(y), r(x); r(y) 2 W ; graph: P | invariance with respect to gauge symmetries depending on time or space u(x) 2 W; u(y) 2 W leads to the transformation rule for connection: P (x; y) ! u(x) 1P (x; y)u(y); | the curvature of a connection P (x; y) is de ned as the conjugacy class of the holonomy along a cycle of a graph: P (x1; x2; : : : ; xk ) = P (x1; x2)P (x2; x3) : : : P (xk ; x1) (the conjugacy means that P 0(x1; : : : ; xk ) e u 1P (x1; : : : ; xk)u for any u 2 W ); the curvature of trivial connection is obviously trivial: P (x1 ; : : : ; xk ) 1; | the gauge principle does not tell us anything about the evolution of the connection itself, so a gauge invariant relation describing the dynamics of connection (gauge eld ) should be added. Standard continuous gauge theories can easily be deduced from the above description by expansion of the parallel transport P (x; y) for two closely situated points x and x + x in continuous space with taking into account that P (x; x) = 1. This leads to the introduction of a Lie algebra valued 1-form A: P (x; x + x) 1 + A x. The curvature 2-form F = dA + A ^ A] is interpreted as a physical strength eld. Finally, Fig. 4. Aharonov{Bohm e ect. Magnetic ux is con ned within one should write dynamical equations for the perfectly shielded solenoid; interference pattern is shifted in the gauge elds. The most important exam- spite of absence of electromagnetic forces acting on the particles ple of gauge dynamics is the Yang{Mil ls theory with the equations of motion dF + A ^ F ] = 0 ; (4) d ? F + A ^ ?F ] = 0 : Note that (4) is a priori (empty) statement called the Bianci identity. Note also that the Yang{Mills equations for the Abelian gauge group U(1) are the same as Maxwell's equations. 7. Gauge connection and quantization. The Aharonov{Bohm e ect (Fig. 4) is one of the most striking illustrations of interplay between quantum behavior and gauge connection. Charged particles moving through the region containing a perfectly shielded thin solenoid produce di erent interference patterns on a screen, depending on whether the solenoid is turned on or o . There is no electromagnetic force acting on the particles, but the working solenoid produces the U(1)-connection, adding or subtracting the phases of the particles and thus changing the interference pattern. In the discrete time, Feynman's path amplitude 6] is decomposed into the product of elements of the fundamental representation of the group = U(1): A
U(1)

= exp(iS ) = exp i L dt

Z

! eiL0;1 : : : eiL

t 1;t

: : : eiLT 1;T :

(5)

By the notation Lt 1;t we emphasize that the Lagrangian is in fact a function de ned on pairs of points (graph edges) | this is compatible with physics where the typical Lagrangians depend on the rst-order derivatives. Thus, the expression P (t 1; t) = eiLt 1;t 2 U(1) can be interpreted as the U(1)-parallel transport. A natural generalization of this is to suppose that | the group may di er from U(1), | the dimension of the unitary representation ( ) may di er from 1. We can introduce a quantum mechanical description of a discrete system interpreting the states 2 as basis elements of a Hilbert space . This allows one to describe the statistics of observations of ; s in terms of the inner product in .


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Now let us replace expression (5) for Feynman's path amplitude by the following parallel transport along the path A ( ) = ( 0;1) : : : ( t 1;t) : : : ( T 1;T ) . Here t 1;t are the elements of a nite group | we shall call a quantizing group | and is a unitary representation of on the space . Recall that all linear representations of nite groups are (equivalent to) unitary and all their characters and eigenvalues are elements of the ring A of algebraic integers 5 7]. It is not di cult to show 8] that algebraic integers are su cient for all our computations (except for normalization of probabilities requiring the quotient eld of the ring A ). With our approach, thus, the quantization becomes a completely constructive procedure. On the other hand, the standard Feynman's quantization can be approximated within our approach by taking the 1-dimensional representations of su ciently large nite groups. 8. A simple model induced by a free particle. In quantum mechanics, as is clear from the never vanishing expression exp i S for the path amplitude, the transitions from one to any other state are possible ~ in principle. However, we shall consider a computationally more tractable models with restricted sets of possible transitions. Let us consider quantization of a free particle moving in one dimension. Such a particle is described by the Lagrangian L = mx2 =2: Keeping only the transitions to the closest points in the discretized space, we come to _ the following rule for the one-time-step transition amplitudes: eimf(x eimf(x
+1) xg2 =(2~) (2~)

= eim= = eim=

(2~)

, :

eim(x x)2 =

= 1,
(2~)

1) xg2 =(2~)

In other words, we have the evolution rule as a U(1)-valued function R de ned on pairs of points (graph edges). Symbolically: R(x ! x) = 1 2 U(1) ; R(x ! x 1) = R(x ! x + 1) = w = eim=(2~) 2 U(1) : (6) Now let us assume that w in (6) is an element of some representation of a nite group: w = ( ), 2 = f 0 = 1; : : : ; M 1g. Rearranging multinomial coe cients (trinomial in this particular case) it is not di cult to write the sum t X ! t! w . Note amplitude over all paths of the form (0; 0) ! (x; t): At (w) = x x + x !(t )! =0 2!2! that x must lie in the limits determined by t: x 2 t; t] : One of the most expressive peculiarities of quantum-mechanical behavior is the destructive interference | the cancellation of non-zero amplitudes attached to di erent paths converging to the same point. By construction, the sum of amplitudes in our model is a function A(w) depending on distribution of sources of the particles, their initial phases, gauge elds acting along the paths, restrictions | like, e.g., \slits" | imposed on possible paths, etc. In the case of the one-dimensional representation, the function A(w) is a polynomial with algebraic integer coe cients and w is a root of unity. Thus, the condition for destructive interference can be expressed by the system of polynomial equations: A(w) = 0 and wM = 1. To be speci c, let us consider the cyclic group = Z = f 0 ; : : : ; k ; : : : ; M 1 g. Any of its M irreducible representations takes the form ( k ) = wk , where w M is one of the M th roots of unity. For simplicity, let w be the primitive root : w = e2 i=M : Figure 5 shows all possible transitions (with their amplitudes) from the point x in three time steps. We see that the polynomial A3 1 = 3w + 3w3 = 3w(w2 + 1) contains the cyclotomic polynomial 4 (w) = w2 + 1 as a factor. The smallest group associated to 4 (w) and hence providing the destructive interference is Z . Thus, 4 Z is the natural quantizing group for the model under consideration. 4 Figure 6 shows interference patterns | normalized squared amplitudes (\probabilities") | from two sources placed in the positions x = 4 and x = 4 for 20 time steps. The left and right graphs show interference pattern when sources are in the same ( = 0) and in the opposite ( = ) phases, respectively. 9. Local quantum models on regular graphs. The above model | with quantum transitions allowed 5 The ring A consists of the ro ots of monic p olynomials with integer co e cients 3].


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Fig. 5. Amplitudes for all possible paths in three time steps

Fig. 6. Group

Z. 4

Interference from two sources at points 4 and 4. Number of time steps T = 20. Phase di erences = 4 4 between sources are 0 and

only within the neighborhood of a vertex of a 1dimensional lattice | can easily be generalized to an arbitrary regular graph. Our de nition of a local quantum model on k-valent graph includes the following. 1. The space X = fx1; : : : ; xN g is a k valent graph. 2. The set of local transitions Ei = fe0i; e1i ; : : : ; ekig is the set of the k edges emi = (xi ! xmi ) adjacent to the vertex xi ; this set is completed by the edge e0i = (xi ! Fig. 7 Fig. 8 xi). 3. We assume that the space symmetry group G = Aut(X ) acts transitively on the set fE1; : : : ; EN g. 4. Gloc = StabG (xi ) 6 G is a stabilizer of xi. 5. i = f!0;i; !1;i; : : : ; !h;ig is the set of orbits of Gi on Ei . 6. the quantizing group is a nite group: = f 0 ; : : : ; M 1g. 7. the evolution rule R is a function on Ei with its values in some representation ( ). The rule R prescribes ( )-weights to the one-time-step transitions from xi to the elements of the neighborhood of xi. From the symmetry considerations, R must be a function on orbits from i, i.e., R(emi g) = R(emi ) for g 2 Gloc. To illustrate these constructions, let us consider the local quantum model on the graph of buckybal l. The incarnations of this 3-valent graph include in particular: | the Caley graph of the icosahedral group Alt(5) (in mathematics); | the molecule C60 (in carbon chemistry). Here the space X = fx1; : : : ; x60g has the shape of Fig. 7 and its symmetry group is G = Aut(X ) = Z Alt(5). 2 The set of local transitions takes the form Ei = fe0i ; e1i; e2i; e3ig, where e0i = (xi ! xi ), e1i = (xi ! x1i), e2i = (xi ! x2i ), e3i = (xi ! x3i) in accordance with Fig. 8. The stabilizer of xi is Gloc = StabG (xi ) = Z . 2 The set of orbits of Gloc on Ei contains the three orbits: i = !0i = fe0i g ; !1i = fe1i ; e2ig ; !2i = fe3ig , i.e., the stabilizer does not move the edges (xi ! xi ) and (xi ! x3i) and swaps (xi ! x1i) and (xi ! x2i): This asymmetry results from the di erent roles the edges play in the structure of the buckyball: (xi ! x1i) and (xi ! x2i ) are edges of a pentagon adjacent to xi , whereas (xi ! x3i) separates two hexagons; in the carbon molecule C60, the edge (xi ! x3i) corresponds to the double bond, whereas the others are the single bonds. The evolution rule takes the form R(xi ! xi ) = ( 0 ); R(xi ! x1i ) = R(xi ! x2i) = ( 1); R(xi ! x3i) = ( 2 ); where 0, 1, and 2 2 . If we take a one-dimensional representation and move 0 | using the gauge invariance | to the identity element of , we see that the rule R depends on the two elements v = ( 1 ) and w = ( 2 ). Thus, the amplitudes in the quantum model on the buckyball take the form A(v; w), depending on two roots of unity. To search nontrivial quantizing groups, one should check (by, e.g., the Grobner basis


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computation) the compatibility of the system of polynomial equations A(v; w) = i (v) = j (w) = 0, where i(v) and j (w) are cyclotomic polynomials. 10. Summary. We proposed an algorithmic approach based on discrete symmetry analysis and implemented in C for the construction and investigation of discrete dynamical models | deterministic, mesoscopic, and quantum. In particular, our approach is applicable to the simulation of nanostructures with nontrivial symmetry properties. Important examples of such nanostructures are (hydro)carbon molecules, like graphenes, fullerenes, etc. We constructed a family of groups unifying the space and internal symmetries in a natural way. This construction generalizes the standard direct and wreath products. We demonstrated that soliton-like moving structures, like \spaceships " in cellular automata, arise inevitably in the deterministic dynamical systems whose symmetry group splits the set of states into a nite number of group orbits. We formulated the gauge principle in the form most suitable for discrete and nite systems. We also proposed a method based on the introduction of gauge connection of a special kind for quantizing discrete systems and constructed simple models for studying the properties of the suggested quantization. We hope that the discrete and nite background allowing comprehensive study may lead to a deeper understanding of the quantum behavior and its connection with symmetries of systems. To study more complicated models, we are developing C programs based on computer algebra and computational group theory methods.

References
1. Poincare H. Mathematics and science: last essays. New York: Dover, 1963. 75{76. 2. Holt D.F., Eick B., O'Brien E.A. Handbook of computational group theory. London: Chapman & Hall/CRC Press, 2005. 3. Kiril lov A.A. Elements of the theory of representations. Berlin{New York: Springer-Verlag, 1976. 4. Kornyak V.V. Discrete dynamical systems with symmetries: computer analysis // Programming and Computer Software. 2008. 34, N 2. 84{94. 5. Oeckl R. Discrete gauge theory (from lattices to TQPT). London: Imperial College Press, 2005. 6. Feynman R.P., Hibbs A.R. Quantum mechanics and path integrals. New-York: McGraw-Hill, 1965. 7. Serre J.-P. Linear representations of nite groups. Berlin: Springer-Verlag, 1977. 8. Kornyak V.V. Discrete dynamics: gauge invariance and quantization // Lecture Notes in Computer Science. Berlin: Springer-Verlag, 2009. 5743. 180{194 (http://arxiv.org/abs/0906.0718). Received November 7, 2009