Discrete Symmetries of Dynamics 869

APPENDIX A7. DISCRETE SYMMETRIES OF DYNAMICS 869

1. If g , g G, then g g G. 1 2 ∈ 2 ◦ 1 ∈ 2. The group multiplication is associative: g (g g ) = (g g ) g . 3 ◦ 2 ◦ 1 3 ◦ 2 ◦ 1 3. The group G contains identity element e such that g e = e g = g for every ◦ ◦ element g G. ∈ 4. For every element g G, there exists a unique h == g 1 G such that Appendix A7 ∈ − ∈ h g = g h = e. ◦ ◦

A ﬁnite group is a group with a ﬁnite number of elements Discrete symmetries of dynamics G = e, g2,..., g G , { | |} where G , the number of elements, is the order of the group. | |

Example A7.1 Finite groups: Some finite groups that frequently arise in asic group-theoretic notions are recapitulated here: groups, irreducible rep- applications: resentations, invariants. Our notation follows birdtracks.eu. Cn (also denoted Zn): the cyclic group of order n. B • Dn: the dihedral group of order 2n, rotations and reflections in plane that preserve The key result is the construction of projection operators from invariant ma- • a regular n-gon. trices. The basic idea is simple: a hermitian matrix can be diagonalized. If this S n: the symmetric group of all permutations of n symbols, order n!. matrix is an invariant matrix, it decomposes the reps of the group into direct sums • of lower-dimensional reps. Most of computations to follow implement the spectral decomposition Example A7.2 Cyclic and dihedral groups: The cyclic group Cn SO(2) of order ⊂ M = λ P + λ P + + λrPr , n is generated by one element. For example, this element can be rotation through 2π/n. 1 1 2 2 ··· The dihedral group Dn O(2), n > 2, can be generated by two elements one at least ⊂ which associates with each distinct root λi of invariant matrix M a projection of which must reverse orientation. For example, take σ corresponding to reflection in operator (A7.20): the x-axis. σ2 = e; such operation σ is called an involution. C to rotation through 2π/n, 2 n 2 then Dn = σ, C , and the defining relations are σ = C = e, (Cσ) = e. M λ j1 h i P = − . i λ λ Yj,i i j − Groups are defined and classified as abstract objects by their multiplication tables (for finite groups) or Lie algebras (for Lie groups). What concerns us in Sects. A7.3 and A7.4 develop Fourier analysis as an application of the general applications is their action as groups of transformations on a given space, usually theory of invariance groups and their representations. a vector space (see appendix A4.1), but sometimes an affine space, or a more general manifold . M A7.1 Preliminaries and definitions Repeated index summation. Throughout this text, the repeated pairs of upper/lower indices are always summed over (A. Wirzba and P. Cvitanović) n b b Ga xb Ga xb , (A7.1) We define group, representation, symmetry of a dynamical system, and invariance. ≡ Xb=1 unless explicitly stated otherwise. Group axioms. A group G is a set of elements g1, g2, g3,... for which compo- sition or group multiplication g g (which we often abbreviate as g g ) of any 2 ◦ 1 2 1 two elements satisfies the following conditions:

868 appendSymm - 11apr2015 ChaosBook.org version15.9, Jun 24 2017 ∈ by g 871 on a ¯ g V (A7.9) (A7.6) (A7.8) (A7.7) G (A7.12) (A7.11) (A7.13) (A7.10) defining ∈ indicates , with the g † V ] matrix n × , and ative ordering of berations. Along n † -dimensional vec- if for any transfor- g space vectors, resp. , the matrix ndices; transposition to say the initial, “el- G = ChaosBook.org version15.9, Jun 24 2017 es: ∈ 1 dual n − g . The action of g G : with all transformations † g given by an [ will always denote the will almost always be assumed and using the unitary condition V V G comes the e ∈ invariant vector b V x g if its elements satisfy commute ] matrix n . of the group is an × ¯ V V . , in which case n V is invariant for all n [ ∈ ∈ b ∈ ¯ x x y b q hermitian . a on a vector ,..., ected by complex conjugation and index transpo- b In what follows 2 a M ff G , a . , , M 1 dual rep x ∈ defining rep b . ) ) is e x . We shall denote the corresponding element of n n a = = g = b b x x ¯ V unitary group a a ) b g b g , y d . Multiplying with ) be a group of transformations acting linearly on c , a = as the † The vector x = ∗ M ( ¯ G ,..., b ,..., ) a M . 2 2 a d ( a b x b x M g a b ) b . , , g † Let = g b c 1 1 0 M ( g x a x x ( a b ( ( g b b . = is given by the = =

A7. DISCRETE SYMMETRIES OF DYNAMICS a x ] G g a b ¯ = b M V gq ) a = = = ∈ b † M ) = ∈ a , † = a g ′ a ′ g M g q ¯ M x x x x [ q ( ( invariant matrix : -dimensional complex vector representation space, that is We shall refer to is an If a bilinear form mation appendSymm - 11apr2015 (A7.9), we find that the invariant matrices Invariant vectors. For a hermitian matrixindices, there as is no need to keep track of the rel sition: Complex conjugation interchangesreverses upper their and order. lower i A matrix is ementary multiplet” space within which we commence our deli with the defining vector representation space tor representation space Hermitian conjugation n Defining rep. to be a subgroup of the Defining space, dual space. hermitian conjugation: In the applications considered here, the group action of a group element raising the index, as indual (A7.3), vectors, so are the distinguished components by of lower, defining resp. upper indic vector ¯

APPENDIX G 870 dual is the ) acts er this (A7.2) (A7.4) (A7.5) (A7.3) F ¯ V , ’s trans- n a ( x dual basis standard rep GL on which these , where the F as a ⊤ ), e distinguished by e ∈ e indices, g dual space C † i g x notation; that interpre- or ChaosBook.org version15.9, Jun 24 2017 , = † ⊤ R ) can be distinguished from ), the row of basis vectors such that n , and zero otherwise. The ′⊤ F x g ) b j e , is spanned by the . This observation motivates ( . n ). g e ] matrices ( ¯ = V n C , and the column of ) be the group of general linear , 1 ,..., × a F is either n 2 − GL , ( n , the space on which x g F n . ∈ , ¯ ( V ( e 1

, then 1 if GL x g . F 0 = V F GL . Under left multiplication the column = , ′ gx e ) linear forms b a transforms as g δ Let n ⊤ = e . -tuples ( ′ a n det ( x b | is mapped into another basis set by multiplication g n ) as V F F → , over the field n b a ( → , the set of ) V } † n ) representation space g d with entries in field GL . ( ( F is a (right) basis of e . a : g . , b is a vector representation space, then the ) , } b standard representation space V ) ) n g j . i y 1 d dual y . We shall refer to the set of [ † V ( δ is the transpose of the inverse of a − ··· e is what it is, regardless of a particular choice of basis, und g , b b = g , ) , x ⊤ If = ( ) a basis set of b x ) † ) b b a ) F (1) 1 j g a A7. DISCRETE SYMMETRIES OF DYNAMICS F e ,..., g , e ( ( ] matrix ) no complex conjugation is implied by the − , ··· g 2 { n e n g F ), and the space of all , n y = ( is the Kronecker symbol, ( · = ( , → , = F × a ) n (1) 1 b a , a i n ′ a b a ( ′ ′ ( = e y δ n GL x { e y g e ( GL ( † GL Under a general linear transformation If by meticulously keeping track of the relative ordering of th GL † transformations, General linear transformations. As the vector Under forms by left multiplication as with a [ transformation its coordinates must transform as Standard rep. transforms by right multiplication as (row transposed) of basis vectors of matrices act as the rep g For g introduction of a tation applies only to unitary subgroups of They transform under Dual space. via the dual rep appendSymm - 11apr2015 where components of dual representationupper indices space vectors will here b set of all linear forms on APPENDIX (left basis) 873 is the factor | i (A7.15) of irreps . Proper- K ) | µ is changed, ( erent classes of the group nd D V ff ), with the new G g ( ′ which simultane- ⊂ conventionally ar- follows: D S H ChaosBook.org version15.9, Jun 24 2017 ) equivalent to the rep- esentation of the ch group element sounds me G ( ′ . A representation of a group D µν . . δ | ) must satisfy a set of orthogo- i j g δ 1 G | | ( | | i − ) G K µ G | | | ( C = ) ∗ D g = = ) ( ν . ( i ∗ 2 µ -dimensional vector space ) erent. Because of the cylic nature of the ) are related by an equivalence transfor- χ d d µ j ) lm ( g ff CD 1 µ ( δ χ ( i C = ) i j r µ χ D µ . This weight comes from the fact that ele- tr | δ ), and we call it the character of ( i that are mapped to the unit matrix of a given i i P µ χ Finding a transformation = µν K ( | G r µ . The equivalent representations have the same δ ) χ r i g G ∈ P = ( P ′ as ) h ) , a square array whose rows represent di ) form a representation 1 D µ g − ( il . ( tr g ′ D 1 ( ) − = D ν ( m j ) C g D ) ( ) g ′ ii ( g ( D ) ) have to be replaced by their transformations µ ) of the group ( il 1 g n CD ( = G i ) and the old matrices D X ( character table g A7. DISCRETE SYMMETRIES OF DYNAMICS D = D ( . = g ′ ) X ) g . This property can be used to define a subgroup H D | ( g 11 / ′ µ ( G d | χ D Here, the summation goes through all classes of this group, a ments in the same class have the same character. number of elements in class ), then according to Schur’s lemmas, = g ( cult. However, suppose it can be achieved, and we obtain a set ) ) 1. The number of irreps is the same2. as the Dimensions number of of irreps classes. satisfy 4. orthonormal relation II : 3. orthonormal relation I : h consisting of all elements µ ffi ( ( D nality rations: ously block-diagonalizes the regular representation ofdi ea Definition: Character tables. resentation structure, although the matrices look di The group of matrices mation through a non-singular matrix trace the character of equivalent representations is the sa matrices the matrices Denote the trace of irrep Equivalent representations, equivalence classes. is by no means unique. If the basis in the The characters for allranged classes into a and irreps of a finite group are group G ties of irreps can be derived from (A7.15), and we list them as appendSymm - 11apr2015 D representation. Then the representation is a faithful repr

APPENDIX G ◦ ∈ = G . 2 ) . 2 ) and and j 872 g g ¯ 1 V ( x ( 11 − V i j D g D ⊗ ) ( (A7.14) = 3 g ( D G homeo- ) V ) is an iso- e D ( ∈ , i.e., in such 1 G ( D = V h d-dimensional j ). d of the group D 1 g P vectors in ( ix multiplication: -dimensional rep- on p = d D ′ i r representations. ) G x in order to distinguish 2 lication, are preserved: ChaosBook.org version15.9, Jun 24 2017 g ( ) of a t do not have to be linear, D g ( ) µ 3 . χ ). g representation d ‘linear’ . In this case the order of the ( of the group. In general, how- G ( ≤ D | j D G | , = i ∈ is deﬁned as trace If the mapping faithful ) ) ≤ that will be mapped on the unit matrix g G 1 ( g ∈ G ( D ) acting on the vector space , is mapped onto the matrix product g D ∈ c G Let us now map the abstract group for 1 ) ( z 2 is mapped onto the inverse matrix G h d i j D is mapped onto the unit matrix g r The character of ( δ . , we say the representation is an ∈ e G ) s d D G = 1 are components of invariant tensor g a ∈ ) a linear or matrix ( ) g y is ) ∈ ii 3 b . We emphasize here e G 1 ◦ cde ( x g ( (repeated indices, as always, summed over). In this V e D − ( i j 2 g D ab p 1 D g cde d h D = ¯ i V X ). ab = corresponds to the matrix operation ) is equal to the order ). g h ⊗ = 1 ( G )) 1 g ( q ) gx 1 = ( − g g D cients V ) ( , since D = ¯ D s is mapped to a matrix ) ◦ d ﬃ D ′ , 2 in ¯ 2 r ≡ ) of the group element x G g = tr , which can be written as a homogeneous polynomial in terms of , g . We will often abbreviate the notation by writing matrices g ( We shall refer to an invariant relation between 1 ¯ ( z ( ¯ ) ∈ V − A7. DISCRETE SYMMETRIES OF DYNAMICS on a group of matrices , = D e ◦ D y ( g )] ) , 3 = χ , i.e., g g x ( g ( ) g ( ( µ 1 D H representation space V χ the inverse element [ g D invariant ) as If the dimensionality of 3. The associativity follows from the associativity of matr 1. Any 4. The identity element 2. The group product G vectors in ( group of matrices morphism, the representation is said to be appendSymm - 11apr2015 ever, there will be several elements Faithful representations, factor group. Note that in the the matrix representations from other representationsin tha general. Throughout this appendix we only consider linea resentation vector components, such as q Invariants. as an example, the coe We call this matrix group a way that the group properties, especially the group multip representation Character of a representation. Matrix representation of a group. morphically APPENDIX D ) 875 d with unitary (A7.19) (A7.18) secular such that .                                C , (or are replaced .                                       2 . with exception . . λ . trix .                                               what follows, we . 0 . ..., . 0 ) . ... 2 1 0 0 λ 3 ChaosBook.org version15.9, Jun 24 2017 3 λ − . . . λ 0 3 − characteristic λ M 2 )( 2 . . . 0 1 λ 2 − λ 0 . 3 . − λ ...... λ 1 0 0 0 10 0 0 1 0 0 M 2                                0 λ ) j 0 λ 2 . . . . 0 0 − λ . . 1 the eigenvalues corresponding to λ ( † 1 0 C 1 , ) 0 j Y 1 2 = λ . distinct roots of the minimal 2 . . † 0 0 0 . λ ... λ ... − 0 r C ) − = M 1 1 j 1 ( 0 ) λ λ λ 1 C i                                               ) factor, has nonzero entries only in the subspace associate − λ 1 2 are the = 1 − λ j

A7. DISCRETE SYMMETRIES OF DYNAMICS M λ † ( λ − 1 M − ( 1 , j , Y 1 λ r M = i a hermitian matrix, there exists a diagonalizing unitary ma i C Y                                       CMC λ M In the matrix It is easy to show that any rep of a finite group can be brought to : 1 λ of the ( and so on, so the product over all factors ( polynomial by zeroes: appendSymm - 11apr2015 Here form, and the same is true of all compact Lie groups. Hence, in APPENDIX specialize to unitary and hermitian matrices. A7.2.1 Projection operators For 874 . We (A7.17) (A7.16) is always } e { ng them e from Hermann termine the num- ries, use properties ause the symmetric . ”: ry function onto the of the corresponding ) tor is known after we provided the diagonal the matrix itself. But µ we need to know the l do not know how to ( i ) n of class rity basis exists or not; ChaosBook.org version15.9, Jun 24 2017 µ ρ ( = D ) x ( ), which is the basis of the 1-d ρ ) gx µ ( ( i ρ P g ; rather, it will be our chief task to P G 1 basis of irrep which takes a regular representation into − | th i rmative in the most important cases.” S G | We have listed the properties of irreps and ffi is known. ) µ ( ii ) D ) g g erent irreps. ( ( ff U U ) ) erent irreps (as usual, 50% of authors, including Chaos- , it is (25.38). But how to get others? We need to resort g . We will use this operator to split the trace of evolution 3 g ( ff ) ( ) µ ) in order to get the bases of invariant subspaces. ( µ µ ( ii ) ( D µ D ( ii χ D . For C g g X X A | | in (A7.16) gives µ µ i G G d d | |

A7. DISCRETE SYMMETRIES OF DYNAMICS = = ) ) µ µ ( ( i P P All invariants are expressible in terms of a finite number amo “ Summing For 1-dimensional representations, this projection opera One of these invariant subspace is investigate for each particular groupthe whether answer, a to finite be integ sure, will turn out a This is also asum projection of basis operator of which irrep projects an arbitra A7.2 Invariants and reducibility What follows is aWeyl bit on the dry, “so-called so first we main theorem start of with invariant a theory motivational quot operator into sum over all di appendSymm - 11apr2015 obtain the character table,for since character 2-dimensional of or 1-d higherdiagonal matrix elements dimensional is representations, It projects an arbitrary function into the cannot claim its validity for every group a block-diagonal form. elements of this representation symmetric irrep to the projection operator: and columns represent di Book, will use columns and rowsber instead). of Rules irreps 1 and and their 2 dimensions. help de As matrix representatio the identity matrix, the firstrepresentation. column is always All the entries dimension ofirrep the is always first 1-dimensional. row3, are To 4 always compute and 1, the the remaining bec class multiplication ent tables. Definition: Projection operators. the techniques of constructing character table, but we stil APPENDIX construct the similarity transformation , ? i q i V 877 P (A7.29) (A7.32) (A7.31) (A7.30) (A7.33) (A7.28) (A7.27) are poly- can be used tional to the M 1 ] matrix reps: commute with : i i M d 2 into a direct sum distinct eigenval- P ices do not com- × r M i A rep is said to be V d is proportional to the ChaosBook.org version15.9, Jun 24 2017 iant matrices commute: with consisting of vectors M i V . : . i commutes with group transfor- i has a block diagonal form: G C i g constructed from ∈ G M be used to further decompose g i i i g X P , constructed from C 1 j = ,... i . M (if nontrivial) can be used to decompose P 3 i ) X ) i . P i M ( 2 : = 0 , i G i 2 M ] hermitian matrix g V = -dimensional vector space P : d 2 M ] d j × i . d M P X r , follows from the orthogonality of , ) -dimensional subspace V i = i ( 2 1            j d (no sum on ⊕ induces a decomposition only if its diagonalized form . . Can M . M P 0 [ . j M G ... i constructed from λ i P 2 ⊕ , ⊕··· 0 0 j P g i , j 2 , X 2 i P 0. Projection operators (A7.20) constructed from . 0 V X 2 V i 1 = 0 0 0 ⊕ = = P g = 0 ] j M ⊕            i 1 ] 1 ] matrix rep can be written as a direct sum of [ ] 1 2 1 P = , so they also commute with all P V d

A7. DISCRETE SYMMETRIES OF DYNAMICS = M M V G ] × if all invariant matrices that can be constructed are propor = M i , , = † M 1 acts only on the ) , j d i commutes with P M ) i → ( 2 1 = G i , = ) ( 2 g P i i ( 2 M M G V CgC [ V M [ P [ G . In this way an invariant [ M -dimensional vector subspaces V i According to (A7.13), an invariant matrix In the diagonalized rep (A7.21), the matrix An invariant matrix Usually the simplest choices of independent invariant matr subspace. d ∈ i Now the characteristic equation for unit matrix. The rep That nomials in ues induces a decomposition of a V (A7.18) has more than oneunit distinct matrix eigenvalue; otherwise and it commutesirreducible trivially with all group elements. q Hence, a [ mations [ of to project commuting pieces of space appendSymm - 11apr2015 Further decomposition is possible if, and only if, the invar mute. In that case, the projection operators projection operators APPENDIX or, equivalently, if projection operators 876 sub- need i P (A7.20) (A7.21) (A7.24) (A7.25) (A7.23) (A7.22) (A7.26) , i never P ) on the i ucible reps: they λ ( that we ] hermitian matrices f -dimensional vector d orthonormality and d d form of the projection × ever invariant matrices ChaosBook.org version15.9, Jun 24 2017 d t [ subspace: i P projection operator . a on C i                                  λ M th subspace is given by 0 i to decompose the orthogonal . , . 1 . ) . j ) are M ) takes the scalar value i i 0 M P ( f subspace is 0 is deployed in the above only as a pedagogical de- th subspace, and zero elsewhere. For example, the 1 i λ C 1 (no sum on , (no sum on . is the eigenvalue of i i 1 . j j P λ . ) λ , the dimension of the . λ i i , λ : The matrices − P , − i ( j i i f 1 P tr . P λ P . M 1 i completeness relation                                  i j i = = δ i λ † i P ) =

A7. DISCRETE SYMMETRIES OF DYNAMICS , j † C P , and that we have used = i = Y tr ) j i C P i = = = M P 1 ,... i 1 i ( r i = 2 f i P d P MP P X CP M , we ﬁnd convenient, the algebraic relations they satisfy, an 1 Thus we can associate with each distinct root projection operator onto the which acts as identity on the The diagonalization matrix to carry such explicit diagonalization; allM we need are what vice. The whole point of the projector operator formalism is It follows from the characteristicoperator equation (A7.20) (A7.19) that and the Hence, any matrix polynomial completeness of As tr ( space and satisfy the This, of course, is thereduce matrices reason and why “operators” one to wants pure to numbers. work with irred A7.2.2 Irreducible representations Suppose there exist several linearly independent invarian M APPENDIX appendSymm - 11apr2015 · T ∂φ ] 879 dx e is so [ (A7.40) (A7.39) R a multipli- →− ). Its transpose φ a matrix; things 1 2 ing an “operator,” ∂ is · , φ ign is, it is there in is also the inverse, T N φ φ. ] ator , φ ), the stepping operator ChaosBook.org version15.9, Jun 24 2017 1 N dx [ − ··· , φ R , µ 3 σ ··· , φ points a derivative is simply a ∂. , 2 1 2 1 φ − N ( , φ σ around, − 1 = − 1 ∂ φ − µ ( = . σ = ) σφ                          · 1 1 φ ] matrix representation of the lattice deriva- 1 T − − N φ × σ . ( = N , . . i a 1                          φ 1 − . 1 j σ . . φ 0 1 − − . j 1 1 µ ℓ = − σ 1 1 h 1 1 − · − 0 1 − µ T 1 i σ − φ 0 1 1 1 σ 1 1 a − 1

A7. DISCRETE SYMMETRIES OF DYNAMICS                          ] matrix. Continuum field theory is a world in which the lattic a 1 1 0 0 1 = The partial lattice derivative (A7.35) can now be written as −                          a 1 N . ℓ µ = × = φ T = σ µ T σ N ∂ ∂ ∂ h σ = Applied to the lattice configuration In the 1-dimensional case the [ 1 ; on a lattice this amounts to a matrix transposition − translates the configuration the other way, so the transpose σ appendSymm - 11apr2015 cation by a matrix: translates the configuration by one site, fine that it looks smooth tothink us. “matrix.” Whenever For someone finite-dimensional calls spaces someth a linear oper To belabor the obvious: On a finite lattice of tive is: If you are wondering wherediscrete the case “integration at by well. parts” It minus comes s from the identity ∂φ finite [ with ‘1’ in the lower left corner assuring periodicity. get subtler for infinite-dimensional spaces. A7.3.1 Lattice Laplacian In the continuum, integration by parts moves APPENDIX stepping operator acts as a cyclic permutation matrix µ ∈ n µ 878 n ) is a ℓ (A7.37) (A7.36) (A7.38) (A7.35) (A7.34) φ , so it is µ σ ν σ ] matrix glory. positive direc- . lattice. N d what it does. d , = × eriodic in each ˆ d ν N ) ) (a vector -dimensional hyper- rator, by introducing -cubic, and let ˆ σ N x , describe the evolution d µ ( / pper shift’ matrix, ChaosBook.org version15.9, Jun 24 2017 d φ σ Z ( Z steps the particle marches ∈ ∈ -dimensional lattice N d , ℓ , ℓ points in all. For a hyper-cubic lattice derivatives . d 0. A sensible way to approximate ℓ N φ = − N a µ ˆ n σ + 0 limit would do, but this is the simplest ℓ µ φ lattice point lattice point ) evaluated on a → x = = = , ( every point is equally far from the ‘boundary’ ects are equally negligible for all points, and the ℓ ℓ ) a φ                          a a x ff ( erent directions commute, = = φ ff is then . x x . − 0 1 . ) for the 1-dimensional case in its full [ µ a ˆ n σ in the direction . a j , + µ 0 1 ˆ n x so defined is nilpotent, and after + ( periodic lattice ℓ , , be the unit lattice cell vectors pointing along the φ δ σ 0 1 ) ) } x x d = = ( A7. DISCRETE SYMMETRIES OF DYNAMICS ( ˆ n j 0 1 0 0 ℓ φ φ ,                          ℓ ) lattice derivative is the lattice spacing. Each set of values of φ = = is the lattice spacing and there are µ = µ ℓ ℓ ··· a σ ∂ a , φ φ σ ( will play a central role in what follows, it pays to understan 2 ˆ cient to understand the action of (A7.36) on a 1-dimensional n In computer dicretizations, the lattice will be a finite Consider a smooth function Let us write down stepping operator the lattice edge, and nothing is left, σ , 2 steps away, the ‘surface’ e ffi 1 / ff ˆ n N appendSymm - 11apr2015 direction. On a an infinite lattice by a finite one is to replace it by a lattice p o cubic lattice where choice. We can rewritethe the lattice derivative as a linear ope As { is no good, as Writing the finite lattice stepping operator (A7.36) as an ‘u A7.3 Lattice derivatives In order to set up continuum field-theoreticof equations spatial which variations of fields, we need to define possible lattice configuration. Assume the lattice is hyper Anything else with the correct where tions. The lattice the translations in di APPENDIX su , d 1 ) is ] is 881 1 = φ [ 2 , φ S inverse 0 σ (A7.46) he prop- φ , then the t is called ( σ = φ etime it is eval- follows from the perator figuration, nding the scope of p; while the ng one. In statistical reduction to invariant commute, and ChaosBook.org version15.9, Jun 24 2017 matrices, meaning that σ erent directions commute, ariant. We will show how on the matrix polynomial ff and σφ 1 vector space into invariant sub- − φ → M φ (A7.39) on a 2-point lattice φ. · σ is 1 φ. − σ into a form suitable to explicit evaluation. and its inverse, M σ · 1 M operator, connecting every lattice site to any other σ − T cient to understand the spectrum of the 1-dimensional φ M , 1 2 ffi T 0 − σ = = T ect of a lattice translation ] φ non-local 1) . ff contributions look like is now clear; as we include higher an φ 1 2 can be used to decompose the [ erential operator connecting only the nearest neighbors, t − # − S ff σ is the (bare) 2-point correlation. In quantum field theory, i σ ··· , so it is su = = µ A7. DISCRETE SYMMETRIES OF DYNAMICS , M 4 0 1 1 0 ] ] 1)( σ " ν + , σφ σφ is constructed from = σ [ [ 3 σ 1 S S σ = ( − ν Consider the e These matrices can be evaluated as is, on the lattice, and som M σ µ uated this way, but in case at hand a wonderful simplification a propagator. observation that the lattice actionthis is works in translationally sect. inv A7.4. A7.4 Periodic lattices Our task now is to transform agator is integral, lattice site. Due toeach the successive row periodicity, is these a aremechanics, one-step all cyclic Toeplitz shift of the precedi invariant under translations, propagator is di higher powers of the Laplacian, the propagator matrix fills u As exchange repeated twice brings us back to the original con to permute the two lattice sites stepping operator (A7.39). To develop asubspaces feeling for works how in this practice,our let deliberations us to continue a humbly, lattice by consisting of expa 2 points. A7.4.1 A 2-point lattice diagonalized The action of the stepping operator σ As the characteristic polynomial of appendSymm - 11apr2015 eigenvalues of What spaces. For a hyper-cubic lattice the translations in di APPENDIX If a function defined on a vector space commutes with a linear o 880 . To 2 in the erence m (A7.44) (A7.42) (A7.45) (A7.41) (A7.43) 2 ff ! ) across three µ ℓ σ φ itten as + d 1 evaluate − µ o compute is to start ciently large σ abola restricted to the me to grips with the ( ffi ChaosBook.org version15.9, Jun 24 2017 2 1 − ] matrix representation of 1 N

× ∂ 1 . d = T N µ X ∂                          4 4 2 − 1 2 a − − = − . 4 6 =                          . You can easily check that it does what − 2 1 1 − . − . . k µ ℓ . . a σ . 4 1 k . non-local 2 = . . While the the Laplacian is a simple di − 1 ) is defined by the value of the continuum function 1 m ℓ ℓ M as a power series in the Laplacian x a 0 − 4 4 1 1 ( = ∞ 2 1 ) 1 − − M k φ 1 X − − µ 1 2 d σ 1 . We shall show below that this Laplacian has the correct 4 6 4 1 1 2 m 1 2 1 1 − − 1 . − − d = = µ X N 4 1 1 4 6 ( ), where 1 6 2 h 1 6 2 1 1 1 s 2 ℓ − − a 1 1 1 2 a −                          − a 1 1 1 a                          ( − 2 4 − φ A7. DISCRETE SYMMETRIES OF DYNAMICS 1 2 a m 1 a 1 = at the lattice point = = = = ℓ = 2 2 x φ M ∆= lattice Laplacian is a finite matrix, the expansion is convergent for su = The Euclidean free scalar particle propagator can thus be wr ) x ( As The lattice Laplacian measures the second variation of a fiel φ the second derivative is supposed to dolattice, by applying it to a par operator (A7.42), the propagator isexpanding a the messier propagator object. A way t 1-dimensional case: neighboring sites: it is spatially appendSymm - 11apr2015 get a feeling for what is involved in evaluating such series, The symmetric (self-adjoint) combination continuum limit. In the 1-dimensional case the [ is the A7.3.2 Inverting the Laplacian Evaluation of perturbative corrections“free” requires or that “bare” we propagator co APPENDIX the lattice Laplacian is: 883 (A7.53) (A7.52) (A7.55) (A7.54) (A7.56) (A7.57) is normalized as k                            P ver, in case at hand f the content of the t way to fritter one’s k 1) k k tions of the eigenvalue k − ChaosBook.org version15.9, Jun 24 2017 2 3 . . . s complete, 1 N ω ω ω ( ω                            . ) N k 1 k √ 1) component of an arbitrary vector k ) ensures that − j j ω N Kronecker delta function for a peri- ( λ ϕ − = be normalized complex unit vectors −                            k k ,ω k ϕ a projection operator that projects a con- λ 1) ( k k (no sum on . k , − k 2 3 σ . . . ··· 1 ) , N ω σ , j ω ω k ( k . The above product kills everything but the yields the 2 ω Q j                            − ,                          ··· ϕ 1 · ,ω + k † k -th eigenvector − j j = . ϕ . 0 1 ϕ . j ,ω 1 k ˜ 1 φ (no sum on . 1 0 ϕ − k = j 1 + k X N j N ω λ 1 -th eigenvalue of λ √ 1 k N 0 1 , − ··· k -th eigenvector of , and the factor − ) kills the k k k P , k j 1 = = λ j σ ϕ δ 0 1 1 k j = = λ δ k φ = =

A7. DISCRETE SYMMETRIES OF DYNAMICS , j onto − 1 0 0 1 j = k † k Y k factor is chosen in order that                          j ϕ ϕ ϕ φ P σ k = · · N P N 1 k k † k † k k √ √ P P X σϕ ϕ ϕ / Constructing explicit eigenvectors is usually not a the bes and orthonormal eigen-direction A factor ( by hand: in expansion a projection operator. The set of the projection operators i condition youth away, as choicetheory of is basis in is projection largelythe operators arbitrary, eigenvectors (see are and so appendix all simple A7.2). that o we can Howe construct the solu figuration The eigenvectors are orthonormal associates with the The 1 as the explicit evaluation of APPENDIX odic lattice appendSymm - 11apr2015 882 sites. ) is the (A7.51) (A7.47) (A7.49) (A7.48) (A7.50) 1 ˜ N φ form sep- , 0 , ˜ onto the two φ 0 1 N−1 ). 2 φ N−2 2 appendix A7.2) . 3 steps the lattice is , φ h . 1 4 φ N s, on which the value and its powers. . 5 c lattice with ChaosBook.org version15.9, Jun 24 2017 σ . . eld ( . k # . enables us to reduce the 2- 1 , and the normalized eigen- 1 } . − 1 1 " # − . 1 2 , # = π 1 − N 1 2 √ i 2 e ) σ 1 1 1 ∈ { -th roots of unity 1 1 1 1 1 = φ " − 2 N λ " 1). As we shall now see, ( in number, the space is decomposed into 2 1 − √ − 2 1 0 , N φ = φ, ( = (1 · ) 2 ) , ω + 1 1 σ √ distinct # P 0 σ 1. The symmetrization, antisymmetrization pro- + 1 1 N − " = − = + 1 : ( . ) 1 2 = 1 ( is a number, φ σ 1 1 1 2 1 n √ translates the lattice one step; in k 1 · ˆ 1 2 n , we can project a lattice configuration 1 σ ) 0 ω = 1 σ , λ ˜ 1 φ 1), ˆ are the P = 1 − φ , 1 1 2 + = σ 1 1 = (1 + = λ σ 1 0 √ ( λ 1 2 ˆ 0 n 1 0 1 − 0 − − φ √ P 0 − φ − λ = ˜ 0 1 ( 1 φ k N Y σ + λ − σ = = 0 0 = = A7. DISCRETE SYMMETRIES OF DYNAMICS = P 1 1 n 0, the symmetric and the antisymmetric configurations trans = = = − # = 1 2 N N 1 0 1 φ φ φ P P " σ σ P 0 Each application of In this way the characteristic equation P 1-dimensional subspaces. The general theory (expounded in N appendSymm - 11apr2015 vectors are ˆ As the eigenvalues are all distinct and dimensional lattice configuration to twoof 1-dimensional the one stepping operator 2-site periodic lattice discrete Fourier transform of the fi normalized eigenvectors of Noting that with eigenvalues so the eigenvalues of back in the original configuration jection operators are A7.5 Discrete Fourier transforms Let us generalize this reduction to a 1-dimensional periodi As APPENDIX arately under any linear transformation constructed from , k ˜ k J -th ) 885 k 0 ˜ G acts as into the (A7.63) (A7.64) (A7.62) ontrast to 1 M . . , ˜ k M ˜ J 1 k tr ors of : ) ln tr relation that 1 -dimensional dual 0 − = − ˜ d G ′ = ojection on the ( µ M N kk † k k ˜ ˜ J ChaosBook.org version15.9, Jun 24 2017 π N M 2 matrices into scalars. If ′ ’s and couplings, such as k J kk ) by its Fourier transform X σ ′ δ ′ ℓ , and the matrix k ′ cos ′ ’s, J = ϕ kk φ kk kk · ) δ X 1 δ J † k = k · d µ = ϕ ˜ ′ ( ′ M ℓ † k P ) ! kk ′ ϕ ℓℓ c 2 1 δ † k ) = 2 a th subspace. For example, for the 1- ′ )( ! ϕ ′ k k − ) 1 k − )( ′ ) P , and from the tr ln ′ k ϕ k 2 n k − ′ 0 · ϕ M ω ϕ ! ˜ k · m M · k M + P 1 β π tr on the ( · k N M 2 M by the diagonalized Fourier transformed ( − † k k -dimensional vector in the = ·

· ℓ = ′ ˜ ϕ d ω ′ X M † k † k ( ℓℓ n )( ) ϕ ′ ϕ kk k 2 1 cos ( δ kk

ℓ M ϕ M X ) k · 2 2 ) k ) is a 2 ], has exactly the same form in the conﬁguration and the 2 = 0 a a µ ϕ T J ˜ ( k J [ G ℓℓ , ( ( ℓ Z ′ , then ( = = k M X , = σ k X ··· ′ ) ℓ , ′ ) kk ′ 2 k = X X k k . In fact, any scalar combination of ϕ J is the trace in either representation , ϕ with · ˜ 1 · · M = =

A7. DISCRETE SYMMETRIES OF DYNAMICS k M ( M M · · · det = † k † k M -th subspace the bare propagator is simply a number, and, in c T ϕ ϕ = k k J ( tr ( OK, a dizzying quantity of indices. But what’s the payback? M Going back to the partition function and sticking in the fact commutes M Fourier space. A7.5.2 Lattice Laplacian diagonalized Now use the eigenvalue equation (A7.55) to convert the partition function det and the spatial propagator ( subspace is a multiplication by the scalar where bilinear part of the interaction, we replace the spatial dimensional version of the lattice Laplacian (A7.41) the pr the trace of From this it follows that tr lattice. appendSymm - 11apr2015 In the APPENDIX the mess generated by (A7.44), there is nothing to inverting = ℓ 884 tary k U (A7.59) (A7.61) (A7.58) (A7.60) ), in which N mpute in the ons) and when envectors (A7.55), (mod y is translationally onormality (A7.54) . 1. The form of the transformation, and t with. For example, j ) er transforms are full k k = ˜ ChaosBook.org version15.9, Jun 24 2017 φ = . . k . Hopefully rediscovering U ′ ℓ † k → 0 1 φ N−1 ϕ ) 2 -th subspace is given by φ N−2 ′ k k of . 3 ϕ · ) . (no sum on 4 k ℓ φ M ℓ . · 5 k π † k N 2 . , . i ϕ . According to (A7.57) the matrix ( − k . k ) e k ′ M ϕ ℓ 1 . 0 − − ′ = ℓ ℓ ( , with Jacobian det X N (no sum on kk π X 1 N 2 wherever profitable: i N = e 1 1 = √ ′ configuration on the 1 k N = k , φ P P k ℓ = = ) ℓ P ) ) M k ′ j ′ P φ ℓ k k ϕ ) − · P discrete Fourier transform ( P k ϕ † k unit vectors pointing at a uniform distribution of points on ( = † k k · ′ ϕ π ˜ N 2 ( φ ϕ N † kk i ℓ X ) M e k is a unitary matrix, so the Fourier transform is a linear, uni · 1 0 as the = = = ϕ − UU ℓ = † k ( k k ℓ X N ϕ ˜ π φ N ( 2 ℓ k = i ) A7. DISCRETE SYMMETRIES OF DYNAMICS 1 ˜ N φ ′ e 1M1 = φ ℓℓ N · ′ = ) 1 = √ k k kk factors (they are eigenvalues of the generator of translati k j ˜ P P M δ ( ( M , the projection of the p = · k ˜ ix ℓ φ The projection operators can be expressed in terms of the eig ) e k ϕ ( The completeness (A7.53) follows fromfrom (A7.58), (A7.57). and the orth is the Fourier space representation of case each term in the sum equals 1. (A7.56) as it this way helps you aof little toward understanding why Fouri are they the naturalinvariant). set of basis functions (only if the theor A7.5.1 Fourier transform of the propagator Now insert the identity the complex unit circle; they cancel each other unless The sum is over the The matrix We recognize transformation, invariant function (A7.46) does not change under APPENDIX from the formal pointFourier space of or in view, the it configuration space does that not we started matter ou whether we co appendSymm - 11apr2015 , ) p 1 A 887 axis σ 2 is the = x x x 3 3 3 24 24 24 y ˜ 24 13 y 4 2 4 4 4 2 4 p etric ( σ σ σ σ σ σ e σ h σ σ σ C C C C C C C me cycles ˜ C . Orbit p retic structure diag cycle. For typo- n / σ p ˜ p transformation that ep, the dynamical n v ed by mapping the = 8 by the accumulated 4 r with respect to the C = 1 ChaosBook.org version15.9, Jun 24 2017 g p , m 2 C and a reflection across a diagonal. = π cycle is 2 g p , ) 1213414234312324 ) 12132124 ) 1214232134324143 ) 12142123 a b a b C , together with the ) are related by time reversal, but cannot be = p b characters ), and a degenerate pair of 2-dimensional 12 example worked out above, this yields ˜ p p 1 0122 12131413 1222 1242413134242313 02110212 12432134 12431423 0021 ( 0221 12421424 1122 12313413 0121 ( 00220102 ( 1213 0222 12424313 0001 12121414 0012 1212414134342323 0111 12143234 1112 1234234134124123 0002 12124343 0112 ( 0011 12123434 2 v 1 4 g B , listed in the last column of table A7.1. Our C of sect. A7.6 has been replaced by 0, so that the , ) and ( 1 a B diag σ 13 σ transformations. = 2. In the , v x x 3 ˜ y 24 y 13 13 2 2 4 4 4 4 labeled cycles cycle into an irreducible segment of the p 4 π/ h σ σ σ e σ σ σ σ } C C C C C C C p . The degeneracy of the axis } 2 σ , ) under both types of reflections, or symmetric under one and C Symmetries of four disks on a rectangle. 1 group elements 2 2 , 1,2,3,4 A v 0 { C { . Substituting the 4 E C = A7.2: correspondence between the ternary fundamental domain pri 1 v

A7. DISCRETE SYMMETRIES OF DYNAMICS group has four 1-dimensional representations, either symm 4 g 1 v C 4 g is a rotation by 2 Figure A fundamental domain indicated by the shaded wedge. C g C A7.1: = This symbolic dynamics is closely related to the group-theo The 12 ˜ p p 011 121434 01 1214 012 121323 02 1243 1 1234 0 12 002 121343 12 12413423 021 124324 2 13 001 121232343414 022112 124213 122 123 124231342413 1 mapped into each other by The two pairs of cycles marked by ( sole boundaryorbit, invariant both under a rotation by where Table of the dynamics: thefundamental global domain 4-disk trajectories trajectory ontoproduct can the of be full the generat 4-disk space ternary alphabet is g convention is to multiply groupsymbol elements sequence. in We need the these reverse groupzeta orde elements function for factorizations. our next st or antisymmetric ( antisymmetric under the other ( representations and the full 4-disk maps the end point of the ˜ appendSymm - 11apr2015 APPENDIX graphical convenience, the symbol 1 x 13 24 } - N 4 , and 886 3 2 1 , Nv 2 mod , i C 1 , and the ǫ 4 symbols); 12 will be C 24 − y 1 ∈ { , the two re- σ is new sym- 1 3 e i 4 + , i ǫ C mple, consider ǫ 13 he vertices of a 2 112 is thus 3 in C tity σ = = i ts until the next re- visitation sequence 3 4 amental domain. In ρ he cycle 3 C he group to 4 disk 2 segment as the d less than 5 are listed to the following 4-disk ChaosBook.org version15.9, Jun 24 2017 2. We start by exploiting → er only by a reflection. As π/ ff di 1 2 1 } i and 3 ǫ π − 2, N { π/ ↔ ↔ ↔ 121323 (note that the fundamental domain and . By reflection across diagonals, an incre- } = } 3 i symmetry, and the disk visitation sequence is ǫ , 1 { 2 2 N + , , only the relative increments C } by angles 3 1 4-disk12 reduced 13 1234 1 3 4 axes, the two diagonal reflections + ... C 2 ∈ { y 3 1 , ǫ i − x 2 ρ 3 ǫ and 2 1 − ǫ 2 Symmetries of four disks on a square. A { 1 . Our convention will be to first perform the increment and C 1 123, where the subscript indicates the increments (or decre − , across 4 2 = 1 y C factorization A7.1: + σ v A7. DISCRETE SYMMETRIES OF DYNAMICS , ··· 4 x 1 corresponds to a flip in orientation after the second and fifth C 2 σ Figure fundamental domain indicated by the shaded wedge. + -disk arrangement has subgroup symmetry in order to replace the absolute labels 3 1 N 4 + C 2 1 + both the fundamental domain and the full space. Similarly, t matter. Symmetries under reflections across axes increase t starting segment, this symbol1 string is mapped into the disk given by disk labels this time the period inparticular, the the fundamental full domain space fixed points is correspond cycles: twice that of the fund the fundamental domain cycle 112. Taking the disk 1 then to change the orientation due to the reflection. As an exa If an A7.6 flections ments) between neighboring symbols; the period of the cycle symbol 1 mapped into 1 add relations between symbols: Conversions for all periodic orbits ofin reduced table symbol A7.1. perio appendSymm - 11apr2015 a consequence of this reflectionflection increments become and decremen vice versa.square (figure A7.1). Consider four The symmetry equal group disks consists placed of the on iden t three rotations bol will be called 1 ment by 3 is equivalent to an increment by 1 and a reflection; th by relative increments APPENDIX the ) ) ) 2 2 all A 1 and 112 112 112 t 889 2 ...... t t t 2 02022 0 ) ) ) B t 0 00211 t − − − n t 2 2 (A7.69) (A7.68) (A7.67) 2 t ...... + 0021 0021 0021 t t t )(1 )(1 )(1 − ) ) − 2 112 ) reflections. t 2 + − + 2 02 02 2 01 v t 022 022 022 t t t t 4 t t t 1 12 1 1 ) 2 t 2 t t t , where t 2 C )(1 )(1 )(1 00121 2 − + + 1 1 t t − − n − t − − 2 2 + ) + t 0 12 )(1 )(1 )(1 0012 0012 0012 12 12 − n t t t t t t 12 y on the special role ( + t 1122 0 0 ) ) ) ) t ... 1) t t 021 021 021 + − + 01102 t t t + 2 ) ChaosBook.org version15.9, Jun 24 2017 + 01 01 12 01 t − 12122 t t t t − − 0011 ( 112 ) t 2 t t 01 1 + 1 + 2 1 − symbol string. For + 2 does not contribute to t t t t t ) ) )(1 )(1 )(1 2 2 t 2 t )(1 − 2 2 = p 0 t 2 021 021 12 12 2 t − − − − − t t t )) (A7.70) t t − )(1 )(1 )(1 ) 02 02 t 2 t ˜p 0011 0011 0011 − + 2 − t t t + + 1 + − 0 t h 011 011 122 011 012 012 012 + − t ( 01021 t t t t t t t 02 1122 12 − 12 − − 02 t 2 t ( ( ( ( 2 t t t t 2 + t 012 012 1 2 B ( )(1 )(1 2 + + − t t t t 2 )(1 − + + + 01 00112 + − ( ( χ − 0 t )(1 )(1 )(1 t 2 02 02 + + ) ) ) ) t t − t , 01 2 + + ) t )(1 )(1 )(1 02 02 1 − − + 02 02 12 02 t t 1 n − + ) ) ( t t t t 002 12 t − 0002 0002 0002 t − t 0 0 1 0 0 t t t − + 011 011 011 t t t t 1) 12 12 0 t t t t 01012 002 t t t (2 112 ) ) t t )(1 )(1 − 1 1 − + + t − − − − 1 1 2 − ( t t )(1 2 11222 − + + does not change sign under t t − 2 t + t the curvature expansion is given by 01 01 2 00002 0 0 t t t 2 2 t 2 ) 2 − − + t t 01 = t − )(1 )(1 )(1 t 002 002 112 002 t 2 E )(1 )(1 )(1 t t t t 1 + + − ( + ) (2 1 − ) − − t ( ( ( ( t 2 ˜p 112 112 0 − 1 + t t t 0022 0001 002 0001 002 0001 002 − − − − h t 01 01 + ( ( t t t t t t t )(1 )(1 )(1 t t ( ) 2 ) ) ) ) 2 0022 1 1 1 ( ( t + 1 2 2 t + t t t t − − + − + − − − 11112 0 B 1 01 01 02 01 t 2 2 t t + + − 2 t t t t χ + − + 2 2 0 0 0 021 2 0 − 1 1 1 t − − 0 , t t t t t 122 122 ( t t t )(1 )(1 )(1 )(1 )(1 )(1 t t t 0 + − n − − + − − )(1 )(1 )(1 ( + + − + 2 − + − 122 001 0 122 001 0 0 122 001 t t 1) t t t t t t t 2 0 0 0 0011 2 02 t t t t t 002 00222 t − 001 001 012 022 001 t t + + − − − + + + + t t t t t ( 2 12 022 022 + − + + 2 t 2 t (2 t ( t ( ( ( ( + − + + 1 + − 1 + (1 (1 + 1 (1 − (1 (1 (1 − + 1 (1 + (1 (1 = ) ˜p ======

A7. DISCRETE SYMMETRIES OF DYNAMICS h ( 2 1 2 2 A E B A B χ /ζ , /ζ /ζ /ζ 1 1 1 1 1 B are the number of times symbols 0, 1 appear in the ˜ with an odd total number of 0’s and 1’s change sign: 1 p appendSymm - 11apr2015 For the mixed-symmetry subspace In the above we have assumed that strings: and n and played by the boundary orbits: by (A7.65) the orbit The form of the remaining cycle expansions depends cruciall APPENDIX t ) = 2 2 2 4 4 24 ) ) ) ) ) 112 888 p ˜ ˜ ˜ ˜ ˜ p p p p p 2 2 2 ... t t t t t t h ) E 13, − − − + + − denotes 2 2 (A7.65) (A7.66) ) ) 0021 ˜ ˜ p p t )(1 t t the symbol ) diag − 2 + + t 022 ff ) (1 ) (1 ) (1 ) (1 ) (1 σ t 1 ˜ ˜ ˜ ˜ ˜ p p p p p t E t t t t t )(1 − 2 )(1 )(1 nd − degenerate pairs ˜ ˜ l space, with the p p + − + − − B t t ) 2 (axes) and 0 12 )(1 0012 t t − / 12 iagonal orbits 2 ( t ) ) ) − ... x 021 − B t − ) − ChaosBook.org version15.9, Jun 24 2017 12 01 ) (1 ) (1 ) (1 ) (1 ) (1 )(1 σ t t ) ˜ ˜ ˜ ˜ ˜ ˜ p p p p p p )(1 01 2 1 − 2 t t t t t t t t t ˜ )(1 p t can be read o E + 1 )(1 2 t 0 0 t − + + − − B t − − 2 1 )(1 02 are listed in the first column; − − t 0011 − C B − ˜ t 2 p ( 122 011 012 − B h t t t 02 − )(1 )(1 02 t ( ( t = ˜ ˜ p p 1 ) (1 ) (1 ) (1 ) (1 ) (1 ( − t t t ˜ ˜ ˜ ˜ ˜ p p p p p )(1 − − p 0 0 1 t t t t t )(1 2 − ) ) 2 − 01 h B ) t A − − )(1 + + − − − A 12 02 1 t t 12 t 0002 − t 1 0 0 t 011 t t 0 t 2 )(1 )(1 t t ˜ ˜ p p − A − − t t − )(1 − 1 − ) (1 ) (1 ) (1 ) (1 ) (1 2 t ˜ ˜ ˜ ˜ ˜ p p p p p − − A 2 01 t t t t t )(1 t 112 002 t 1 is replaced by 0 in the remainder of this sec- 1 )(1 t t 1 − ( 11 11121 11 11 1 -1 -1 -1 1 1 -1 -2 -1 -1 0 1 0 0 t (1 (1 ( ( A − − − − − A 0 − t 0001 002 − − t t )(1 (1 (1 (1 (1 (1 2 ) ) 1 t + = = 2 fixed point in the fundamental domain. 3 4 t − − v 02 01 cycle expansion is given by C 2 − t t 4 − e , diag axes 2 2 021 2 0 v = = = = = 1 C 4 t t t C ) ) does not change sign under reflections across symmetry 4 t )(1 )(1 σ σ ˜ ˜ p p 2 2 C ˜ p t t C + − − )(1 − t 4 4 2 4 8 0 122 001 t t t ) ) ) ) ) 0 − − t ˜ ˜ ˜ ˜ ˜ p p p p p 2 2 4 2 012 022 001 t t t t t − − − t t t − ( ( ( − − − − − − − 1 (1 − (1 (1 = = A7. DISCRETE SYMMETRIES OF DYNAMICS 2 (diagonals) respectively: : (1 : (1 : (1 : (1 : (1 subspace in ˜ axes: (1 p 3 2 / 4 e 1 h 1 ) C C A A diag axes 13 , 4 /ζ σ σ diagonals: (1 σ denotes a reflection across either the x-axis or the y-axis, a 1 C + The 2 axes C a reflection across aof boundary diagonal orbits (see can figurefundamental run A7.1). domain along group the In theory symmetry weights lines addition, in the ful σ ( The possible irreducible segment group elements (we have assumed that axes). For the 4-disk arrangement considered here only the d (for typographical convenience, 1 into (25.20) we obtain: tion). For 1-dimensional representations, the characters occur; they correspond to the APPENDIX appendSymm - 11apr2015 ) 2 A 112 891 ... t ) + (A7.71) 0021 t )(1 ) + 2 t 022 t 1 cycle. The de- t )(1 + p − ) ) ) ) with an odd total transformation that ˜ ˜ ˜ ˜ p p p p ) t t t t v 12 )(1 p 0012 2 t 2 t t 12 ontributions of peri- ( t + − − − B ) ) C ... 021 + t + ) ChaosBook.org version15.9, Jun 24 2017 + 12 01 transformations. The full oup theory factors from all t t ) ubspace factorization v 01 2 1 − fundamental domain prime 1 2 t 2 t t )(1 t ed code by the rules given in } )(1 2 B 0 C ) (1 ) (1 ) (1 ) (1 t 2 t − − ˜ ˜ ˜ ˜ p p p p )(1 02 , t t t t t 0011 − 1 − 1 t , 122 011 012 − + − − − B 0 t t t 02 − 02 { t ( ( . For t 1 ( v x x x x x − y y y y y 2 2 2 2 2 t 4 )(1 − + e e e σ σ σ σ g σ σ σ σ σ σ C C C C C )(1 − ) ) C − 01 ) t ) (1 ) (1 ) (1 ) (1 )(1 12 02 ˜ ˜ ˜ ˜ 1 p p p p t t 12 t , together with the t t t t 0002 + t 1 0 2 0 t p 011 t t 0 t t t + + − − A 1, and the corresponding cycle expansion − − − − )(1 + . Note that the 012 and 021 cycles are related − + 2 p t 2 01 n )(1 t 112 002 t )(1 / cycles t t 1 + ( ˜ t p ( ( ) (1 ) (1 ) (1 ) (1 } ˜ ˜ ˜ ˜ 0 p p p p n + t t 0001 t 002 t t + + 4 t t 1 )(1 2 ) ) 1 t + − − − − A t + + = factorization of 02 01 + t t − cycle into an irreducible segment of the (1 (1 (1 (1 p 2 1,2,3,4 ˜ 021 2 0 p p 1 0001 14143232 0002 14142323 00110012 1412 14124143 0021 14134142 00220102 1413 14324123 0111 14343212 0112 14342343 0121 14312342 0122 14313213 0211 14212312 0212 14213243 0221 14243242 0222 14242313 1112 12124343 11221222 1213 12424313 { t t t t )(1 )(1 B p m + − − )(1 − = = = = 0 122 001 t t t 0 t 012 022 001 all characters are + − − t t t x x x x y y y y 2 2 2 2 + 2 2 2 4 ( ( ( 1 ) ) ) ) σ σ σ σ σ σ σ e e σ g C C C C cycle is ˜ ˜ ˜ ˜ − + 1 (1 − (1 (1 p p p p 2 2 2 A t t t t p correspondence between the ternary − − − − v = =

A7. DISCRETE SYMMETRIES OF DYNAMICS 2 C 1 B and the full 4-disk : (1 : (1 : (1 : (1 ˜ p y x 2 e /ζ p g A7.2: 1 σ σ C Substituted into the factorized determinant (25.19), the c ˜ p p 02 1423 0 14 1 12 01 1432 12 1243 001 141232 2 13 011 143412 122 124213 012021 143 022 142 142413 002 141323 112 121343 table A7.2. For Cycle expansions follow by substituting cycles and their gr appendSymm - 11apr2015 is given in (A7.66). Similarly, the totally antisymmetric s is given by (A7.67), the number of 0’s and 2’s change sign: maps the end point of the ˜ cycles ˜ by time reversal, but cannot be mapped into each other by odic orbits split as follows Table space orbit listed here is generated from the symmetry reduc sect. A7.7, starting from disk 1. APPENDIX generacy of the exercise 23.8 k 890 their 142) sym- under v = 2 C , we associate cs for 4 suf- z y ain one has to by a reflection bit leaving the + σ 1 = orded by the topo- 143 and 021 e listed in table A7.2. = hese two reflections ff 1 gle. However, if the ctangle has = E g opposed disk cross the nary description is in- ChaosBook.org version15.9, Jun 24 2017 2 /ζ 1 and , i.e., the reflections across = x } 2 2 2 1 1 σ B B C − − , = /ζ y 1 0 , and the associated group-theory 1 1 1 1 g ,σ B x − − , factorization is a symmetries. ... ,σ v 1 2 E 2 2 e 11 1 ζ into the expansion yields representation is symmetric with respect { 2 ce for constructing the covering symbolic = A C − − p B 1 1 n . . ffi ζ z A B 0 π 1 g B 1 = /ζ 11 11 1 1 1 1 0 ζ A the straight and the curved sections of the bil- 1 2 g p A t 1 by any ζ = ff g 1 v 0 x y 2 2 2 A e consists of g A not ζ σ σ C 2 C representation is antisymmetric with respect to both. v g /ζ 2 = 2 0 1 C g A ζ 1 g , z ords a rather easy visualization of the conversion of a 4-dis has four 1-dimensional representations, distinguished by ... 3 , to which we assign the label 2. For example, a ternary string ff v y 2 − is converted into 12143123 σ representations are symmetric under one and antisymmetric factorization C 1 x 2 v

A7. DISCRETE SYMMETRIES OF DYNAMICS σ = B 2 ... 1 = C A 2 and /ζ C 1 1 B cient. For example, in the stadium billiard fundamental dom The group Short ternary cycles and the corresponding 4-disk cycles ar The above is the complete description of the symbolic dynami A quick test of the This system a ffi dynamics. behavior under axis reflections. The the other reflection. The character table is weight is given by to both reflections; the The liard wall; in that case five symbols su Note that already at length three there is a pair of cycles (01 0010201 appendSymm - 11apr2015 distinguish between bounces o labels 1 and 0, respectively. Orbitsboundaries going of to the the fundamental diagonally domainis twice; the just product of t ficiently separated equal disksfundamental placed domain at corners requires of furthersu a partitioning, rectan the ter related by time reversal, but the symmetry axes and a rotation by in agreement with (18.46). A7.7 An arrangement of four identical disks on the vertices of a re metry (figure A7.2 (b)). dynamics into a fundamentalfundamental domain domain symbolic through dynamics. one ofon An the that or axis; axis may with these be symmetry folded operations back APPENDIX logical polynomial; substituting + . P 1) , use − (A7.76) first ? , ρ 4 m − undefined P = , 0 ,π/ . Beyond that, + d M P 1) = − ) ρ α , θ ( ( b . d 0) 2 # , P p a ( (0 From W.G. Harter [26.39] b 2 . + √ a = = − satisfy ) ) ) α gets reduced to a set of subspaces 1 0 c c ( b z z 2 d d , , + 1 0 √ to irreps, then proceed with computing , and a 2-dimensional subspace, with r a ( , θ b − a 1 r " + ( 893 = a ) + = ( , and show that the 3-pendulum system decom- a − 2 ) P ) − 1. Show that (11.12) has two equilibria: ( ω them to reduce (d) Does anything interesting happen if remaining eigenvalues of whose dimensions The point of the above exercisesymmetry is reduction that is almost only always the partial:tion a of matrix representa- dimension The exercise is simpleout enough using that the you symmetry, can so: do it construct with- Lorenz system in polar(continuation coordinates: of exercise dynamics. 11.3) love many, trust few, and paddle your own canoe. (c) Construct the corresponding projectionand operators acceleration matrix (trust yourfrom own what algebra, is if stated here) it strays poses into a( 1-dimensional subspace, with eigenvalue A7.3. ChaosBook.org version15.9, Jun 24 2017 al domain trajectories onto group elements. , imensional anisotropic Kepler v         4 ure of the dynamics: the global C 1 2 3 x x x is . Assume k                 ax − b ness symmetry. c = interchanges the 0 ff + . − While there is a variety of labeling conven- 2 ¨ l x R / C dynamics, we prefer the one introduced here g v b = 4 1. is the symmetry of several systems studied in the a a a + C b − − v , the one midway of same ≪ c l From W.G. Harter [26.39] 2 2 l , where , with the tip coupled to the C } / and 0, i.e., that the dynamics is i M R . x , b         = e c Ml { 0 + / − ] k a = R Show that multiplication table = f cadbe eabcdeabcdcbf f f ead aedbfbdef ca c df caeb ,         2 and length a c − C , m erent mass f = c e b a d 0 00 1 11 0 0 0 symmetry labeling conventions ml ff         / v v         2 4 k 1 2 3 = ¨ ¨ ¨ C C x x x = R         a A7.2 A7.1 describes a group. Or doestable it? satisfies (Hint: the check group whether axioms this of appendix A7.1.) Three coupled pendulums with a tips of the outer ones with springs of sti Consider 3 pendulumsthe in same a mass row: the 2 outer ones of outer pendulums, length but di Am I a group? displacements are small, (a) Show that the acceleration matrix invariant under (b) Check that [ where exerAppSymm - 1feb2008 Remark literature, such as the stadiumpotential [A39.17]. billiard [9.10], and the 2-d the full 4-disk space by the accumulated product of the because of its close relation4-disk to trajectory the can group-theoretic be struct generated by mapping the fundament tions [24.19, 11.14] for the reduced Commentary Remark Exercises A7.2. EXERCISES A7.1. ) 1 = 112 892 ... t 1 the 0 the ) b + − , (A7.72) (A7.74) (A7.73) (A7.75) = 0021 b t )(1 b ) − 2 t 022 t 1 For t )(1 − . For − } on–wandering set diagonal. ) 1 together with their 12 )(1 0012 n onvergence still ap- , tion reversing t t 12 ( x t 1 ) ) ... 021 − − s given by t − ) − ChaosBook.org version15.9, Jun 24 2017 12 01 t t ) symmetry in the param- = 01 2 1 − 2 ∈ {− t t t )(1 t 2 )(1 1 2 0 b t b t + − − / )(1 02 n t 0011 − a x − t 122 011 012 + , t t t 02 − z 02 t ( ( → t 1 ( − t + )(1 + − a )(1 + , ) ) 1 − 01 ) b t )(1 12 02 1 = / t t 12 t 0002 + t 1 1 0 0 t 2 011 t t 0 t t t B 1 + ζ − − − → )(1 − − 2 t in the coordinate plane, and there is no need b 2 = 01 )(1 t 112 002 t )(1 b t t 1 + ( 1 diagonal. This is one of the simplest models of t ( ( . / B 1 0 n x ) + ζ t 0001 002 + + x 1 t t )(1 2 ) ) + 1 t + = t n + + = , , 02 01 x + 2 t t → − 1 + 1 1 A 021 2 0 1 1 + + + − t t t t )(1 )(1 ζ x n n n 2 n x x x + − − )(1 + 0 122 001 t t t ax 0 − + , t 012 022 001 2 n 2 n − + + − t t t − ) parameter plane outside the strip ( ( ( z b ax ax − − 1 (1 + (1 (1 (1 3 , b 1 − − a − 1 − 1 = = 1 A7. DISCRETE SYMMETRIES OF DYNAMICS with an odd total number of 1’s and 2’s change sign: ´ enon map symmetries = = = p = t 2 1 1 1 B 1 − − − A n n n 1 all /ζ ζ x x x 2 1 B The topological polynomial factorizes as Note that all of thepseudo-orbit above shadows, cycle so expansions that group theply. long shadowing arguments orbits for c ńnmapHénon is reversible: the backward iteration of (3.18) i For the backward and the forwardis iteration symmetric across are the the same, and the n consistent with the 4-disk factorization (18.46). A7.8 H We note here a few simple symmetries of the map Hénon (3.17). to explore the ( Hence the time reversal amounts to map is orientation and area preserving , a Poincaréreturn map for a Hamiltonian flow. For the orienta eter plane, together with appendSymm - 11apr2015 and the non–wandering set is symmetric across the APPENDIX case we have (A7.80) -dimensional dual lattice, and N , 17 895 Revs. , 2838 is the lattice spacing. is a site in the k N / Phys. Rev. Chaos A39 L = , 1183 (1999) dynamical sys- where a , 251 (1993). tion in the Pres- erization of the 8 6 bits in dynamical mics,” eries data,” D (Wiley, New York cations of toroidal actors,” e Lorenz attractor,” ts in dynamical sys- , 1 (1983). Phys. Rev. 20 ChaosBook.org version15.9, Jun 24 2017 , ) (Chelsea Publ., New York 1 − (Dover, New York 1961). Nonlinearity N , 47 (1983). / k 22 π Cont. Math. ′ kk δ cos 2 ( 2 ) The topology of chaos 2 ma Topology ( − IEEE Trans. Image Processing 2 (Addison-Wesley, Reading, MA 1971). m = , 259 (1994). ′ k 93 ϕ , 5597 (2007). · Lectures on Linear Algebra 40 Fundamentals of Linear Algebra − , 1455 (1998). 1 1 2 , 6052 (1990). 70 Linear Algebra m · † k A41 th subspace the propagator is simply a number, k J. Phys. A ϕ , 036205 (2003). 67 and, in contrast to theis mess nothing generated to by evaluating: (A7.77), there In the systems I: Lorenz’s equations,” tems II: Knot holders for fibered knots,” Mod. Phys. tems,” E Lorenz attractor by unstable periodic orbits,” 013104 (2007). chaotic attractors,” arXiv:0707.3975. 2002). ence of Symmetry - Part I,” 1979). J. Zeitschr. Physik B (1989); [A7.15] J. Birman and R. F. Williams, “Knotted periodic orbi refsAppSymm - 13jun2008 [A7.14] J. S. Birman and R. F. Williams, “Knotted periodic or [A7.13] R. Gilmore, “Topological analysis of chaotic time s [A7.12] C. Letellier and R. Gilmore “Symmetry groups for 3 [A7.10] R. Gilmore and M. Lefranc, [A7.2] S. Lang, [A7.9] R. Gilmore, “Two-parameter families of strange attr [A7.11] R. Gilmore and C. Letellier, “Dressed symbolic dyna References [A7.1] I. M. Gel’fand, [A7.3] K. Nomizu, [A7.4] J. M. Robbins, S. C. Creagh and R. G. Littlejohn, [A7.6] V. Franceschini, C. Giberti and Z.M. Zheng, “Charact [A7.8] C. Letellier, R. Gilmore and T. Jones, “Peeling bifur [A7.7] B. Lahme and R. Miranda, Decomposi “Karhunen-Loève REFERENCES [A7.5] B. Eckhardt and G. Ott, “Periodic orbit analysis of th - . =                          ) ′ k 4 4 ϕ erence propa- − − · (A7.78) (A7.77) (A7.79) ff matrices is: M . 4 6 σ · 2 6 ′ − † k kk ϕ inverse δ Insert the iden- ! 1 − . 4 1 , then ( ! . . k − . σ π n N 2 4 4 1

n acts as a multiplication by − − with 2 1 m cos M

0 4 6 41 1 1 th subspace. Show that for the = , lower Poincarésection of fig- ∞ 2 − − k n 2 X 0 th subspace is a k 2 . To get a feeling for what is in- 1 2 = 41 1 4 6 m EQ 1 6 ) m commutes − − contributions look like is now clear; as ′ k = on the                          ϕ wherever you profitably can, and use the M ) · 4 ··· k 1 1 ( 894 a , − ˜ erential operator connecting only the nearest 4 = 1 M · = ff 1 ) † k 2 , k 2 ( , and the matrix ϕ 3 ′ m ( P manifold of ure 14.8(b). Elucidate itsreturn relation map to of the figure 14.9. Poincaré (plot by J. Halcrow) tation Lorenz is alsoflow, periodic their orbit Floquet of multipliers the aredo Lorenz the the same. Floquet How multipliers ofbits of relative the periodic representations or- relate to each other? look like now? Interpret. is a finite matrix, the expansion is convergent for kk δ ciently large P ) k 5. Show that if a periodic orbit of the polar represen- 6. What does the volume contraction formula (4.42) ( ffi ˜ M operator (A7.42), its inversestatistical (the mechanics “free” and propagatormessier of object. quantum A field waypropagator to theory) as compute a is is power to series start a in expanding the Laplacian Laplacian is a non-local operator. While the Laplacian is a simple tri-diagonal di Lattice Laplacian diagonalized. tity eigenvalue equation (A7.55) to convert shift into scalars. If the scalar 1-dimensional version of thethe lattice projection on Laplacian the (A7.42) cise A7.5. neighbors, the propagator ising integral every operator, lattice connect- site to any otherThis lattice matrix site. can besometime evaluated it as is is, evaluateda on this wonderful simplification the way, follows lattice, but fromthat the in and observation the case lattice at action hand is translationally invariant, exer What we include higher andthe higher propagator powers matrix of fills thegator up; Laplacian, is while di the As volved in evaluating such series, show that su A7.4. A7.5. ChaosBook.org version15.9, Jun 24 2017 - 2 / 1 C ), 1

+ n 4 s − , 194505. -axis and the n 28. Topolog- . z s , (and its 10 20 1 = 10 × i 8450) ρ . 0), with contract- The stable eigen- ± EQ , 0 3, 0) has eigenvalue − 6163306, / 0 , . , has contraction rate , 3988). 0) 8 n . 0 0 95686 , 059617, S 0 . . = = 1 1 − EQ , 969775 b 093956 2010 . . . . ≈ = (2) 8464 0 0 0 . ) 757316 ) equilibrium: . − -20 ... − 0 10, = 0 , (2) (2) , 3298 1) of − , π/ω . = , , equilibrium: 2 0 27) (2) 0 /ω /ω 666 , . σ − , ω = 0) , (1) i (2) 2 4955 . , (0 ) has one stable real eigenvalue -40

12 − 5325 T 2 0 0 ± 0 . 244001 πµ πµ 854578, 8277. The unstable eigenvector , 20 . = , . . 653049

− =

-20 -40 8277. 0 ( n+1 (0

(2) . 8557 (0 1) are plotted in figure 14.8. (0 S 0 EQ is arc-length measured along the unstable . 13 22 b − (1) µ = = ( − s e + 11 − − − ( (0 = = exp(2 exp(2 = n = (2) (2) = = = = s = = 1 0 = = 3) e e , , n c r (3) (1) (2 (2) (2) (3) (3) (2) s EQ EQ where vectors of thethe two precise numbers equilibria tograms): are help you (we check your list pro- here The unstable eigenplanevectors isRe defined by eigen- Im with period and the contracting multiplier Λ λ Lorenz coordinates figure 2.5, and inpolar the angle doubled- coordinates (11.10) for therameter values Lorenz pa- radial expansion multiplier Λ λ and the unstable complex conjugateλ pair ing eigenvalue λ e rotation e The other stable eigenvector is e vector λ along the stable eigenvector of equilibrium in the doubledtation, polar and the angle corresponding represen- Poincaréreturn map ( ically, does it resembleRössler attractor, the or Lorenz neither? butterfly,tion the of The the Poincarésec- Lorenz flow fixed by the 2. Verify numerically that the eigenvalues and eigen- 3. Plot the Lorenz strange attractor both in the 4. Construct the Poincaréreturn map (

EXERCISES exerAppSymm - 1feb2008 - J. Z. N 896 Vilaseca, V. Optics Comm. in non-Abelian arge-scale struc- ,” f the structure of er and L. Pittner, tte turbulence,” f the classical erences among at- , 026212 (2005). lassified by bound- ff corps et les distances 72 d information flow,” ChaosBook.org version15.9, Jun 24 2017 N ebras as invariance alge- (Springer-Verlag, Berlin , 269 (1996). , 204 (1996). 84 211 0604062. / Phys. Rev. E (2003). , 056206 (2004). 69 (Springer, New York 2000). , 4 (1969). J. Stat. Phys. Phys. Lett. A 10 , 1536 (1976). 14 , 151 (1998). , 134104 (2003) 77 (June 1977, unpublished); available on Chaos- 31 91 / Phys. Rev. E ,” 3 R Proceedings ICMP03 J. Math. Phys. Emergence of the theory of Lie groups: An essay in the Phys. Rev. D , 339 (2007); arXiv:physics 580 Invent. Math. , 80 (1981). refs. Phys. Rev. Lett. / 36a Recent developments in mathematical physics , 367 (1995). Fluid Mech. ing tori,” 117 tractors and their intensity maps in the laser-Lorenz model mutuelles,” body problem,” in tural reorganization of strange attractors,” history of mathematics, 1869-1926 Espinosa and G. L. Lippi, “Structural similarities and di chaotic attractors in Book.org bras,” Oxford preprint 40 1987). Naturf. gauge theories,” eds., refsAppSymm - 13jun2008 [A7.30] D. Viswanath, “Recurrent motions within plane Coue [A7.17] T. D. Tsankov and R. Gilmore, “Topological aspects o [A7.27] A. Albouy and A. Chenciner, “Le des problème [A7.16] T. Tsankov and R. Gilmore, “Strange attractors are c [A7.28] T. Hawkins, [A7.26] A. Chenciner, “Symmetries and ‘simple’ solutions o [A7.18] O. E. Lanford, “Circle mappings,” pp. 1-17 in H. Mitt [A7.25] W. G. Harter, [A7.19] C. Letellier, T. Tsankov, G. Byrne and R. Gilmore, “L [A7.20] R. Shaw, “Strange attractors, chaotic behavior, an [A7.29] N. B. Abraham, U. A. Allen, E. Peterson, A. Vicens, R. [A7.21] D.J. Driebe and G.E. Ordnez, [A7.22] G.E. Ordonez and D.J. Driebe, [A7.24] “Classical P. Cvitanović, and exceptional Lie alg References [A7.23] P. Cvitanović, “Group theory for Feynman diagrams