Structure of Elementary Particles in Non-Archimedean Spacetime

Structure of Elementary Particles in Non-Archimedean Spacetime Jukka Virtanen Department of Mathematics University of California Los Angeles USC Lie Groups, Lie Algebras and their Representations Workshop Los Angeles, May 1-2, 2010 Jukka Virtanen (UCLA) Lie Algebras Workshop 1 / 43

Acknowledgement The results presented in this workshop are from joint work with Professor V. S. Varadarajan (UCLA) V. S. Varadarajan and J. Virtanen, Structure, Classification, and Conformal Symmetry, of Elementary Particles over Non-Archimedean SpaceTime, Letters in Mathematical Physics, (2009), 171-182. V. S. Varadarajan and J. Virtanen, Structure, Classification, and Conformal Symmetry, of Elementary Particles over Non-Archimedean SpaceTime, p-adic numbers, ultrametric analysis and applications Vol. 2 No. 2 2010 V. S. Varadarajan and J. Virtanen, Structure, classification, and conformal symmetry of elementary particles over non-archimedean space-time, arxiv:1002.0047v1 [math-ph], 2010. V. S. Varadarajan, Multipliers for the symmetry groups of p-adic space-time, p-adic Numbers, Ultrametric Analysis, and Applications, 1(2009), 69 78. Jukka Virtanen (UCLA) Lie Algebras Workshop 2 / 43

Introduction 1 Volovich hypothesis. 2 Symmetries of quantum systems. 3 Projective unitary representations and Elementary particles. 4 Multipliers. 5 Ordinary Mackey machine. 6 Variant of Mackey machine for m-representations. 7 Elementary particles of the p-adic Poincaré group. 8 Elementary particles of the p-adic Galilean group. 9 Some remarks on conformal symmetry of p-adic particles. Jukka Virtanen (UCLA) Lie Algebras Workshop 3 / 43

Some prophetic remarks Now it seems that the empirical notions on which the metric determinations of space are based, the concept of a solid body and light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypothesis of geometry; and in fact, one ought to assume this as soon as it permits a simpler way of explaining phenomena. -Bernhard Riemann, Inaugural lecture, 1854 In fact, I would not be too surprised if discrete mod p mathematics and the p-adic numbers would eventually be of use in the building of models for very small phenomena. -Raoul Bott, 1975 Jukka Virtanen (UCLA) Lie Algebras Workshop 4 / 43

Some prophetic remarks Now it seems that the empirical notions on which the metric determinations of space are based, the concept of a solid body and light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypothesis of geometry; and in fact, one ought to assume this as soon as it permits a simpler way of explaining phenomena. -Bernhard Riemann, Inaugural lecture, 1854 In fact, I would not be too surprised if discrete mod p mathematics and the p-adic numbers would eventually be of use in the building of models for very small phenomena. -Raoul Bott, 1975 Jukka Virtanen (UCLA) Lie Algebras Workshop 4 / 43

Motivation Bit of historical introduction Non-Archimedian geometry developed in 19th century to answer questions in number theory. Fields that satisfy the ultrametric inequality: d(x, z) max{d(x, y), d(y, z)}. Most notably the p-adic numbers. Theme developed over years to study well known questions where the underlying structures are over R or C over the new fields. Arithmetic physics is the study of well known algebraic structures in quantum mechanics over new fields and rings. In 1930s Weyl considered quantum mechanics over finite fields. In 1970s Beltrametti and his collaborators started to investigate alternative possibilities for the micro-structure of spacetime based on p-adic fields. Jukka Virtanen (UCLA) Lie Algebras Workshop 5 / 43

Volovich hypothesis How small is small? Planck length (10 33 cm) and Planck time (10 44 s). Planck length is the smallest possible measurable distance and Planck time the smallest measurable time. Planck length arises from interplay between general theory of relativity and quantum mechanics To probe region of space on sub-planck scale one needs energy greater than Planck mass. This will create a miniature black hole. Volovich (1987) Archimedean axiom is at its core a statement about comparison of lengths. At sub-planck scale the Archimedean property breaks down. At Planck scale spacetime itself fluctuates and transitions from R to Q p cannot be ruled out. New geometry of spacetime based on a non-archimedean field. Jukka Virtanen (UCLA) Lie Algebras Workshop 6 / 43

Volovich hypothesis How small is small? Planck length (10 33 cm) and Planck time (10 44 s). Planck length is the smallest possible measurable distance and Planck time the smallest measurable time. Planck length arises from interplay between general theory of relativity and quantum mechanics To probe region of space on sub-planck scale one needs energy greater than Planck mass. This will create a miniature black hole. Volovich (1987) Archimedean axiom is at its core a statement about comparison of lengths. At sub-planck scale the Archimedean property breaks down. At Planck scale spacetime itself fluctuates and transitions from R to Q p cannot be ruled out. New geometry of spacetime based on a non-archimedean field. Jukka Virtanen (UCLA) Lie Algebras Workshop 6 / 43

Main theme of the Talk We want to describe the elementary particles under the assumption that the underlying field is a non-archimedean field. Jukka Virtanen (UCLA) Lie Algebras Workshop 7 / 43

Elementary Particles, Symmetry Groups and Projective Unitary Representations (PUR) Description of a quantum system S Underlying object is a complex separable Hilbert space H. States of the system are points of P(H). We can define the transition probability in P(H) to be: Symmetries of a quantum system S. Definition p([ψ], [φ]) = (ψ, φ) 2, ψ, φ H. A symmetry of a quantum system is a bijection P(H) P(H) that preserves p([ψ], [φ]) for any two [ψ], [φ] P(H). Jukka Virtanen (UCLA) Lie Algebras Workshop 8 / 43

Projective Unitary Representations (PUR) Symmetries are induced by unitary or anti-unitary operators, unique up to phase. R : H H, [φ] [Rφ] As a consequence that the symmetry of a quantum system with respect to a group G may be expressed by a projective unitary representation (PUR) of G. Definition A projective unitary representation of G is a map U of G to the unitary group U of H such that U(1) =1 and U(x)U(y) =m(x, y)u(xy). m is Borel map from G G to T and is called a multiplier for G. We also call U an m-representation of G. Elementary particles of a group G are defined to be its irreducible projective unitary representations (PUIRs). Jukka Virtanen (UCLA) Lie Algebras Workshop 9 / 43

Topological Central Extensions A PUR may be lifted to an ordinary unitary representation (UR) of a suitable topological central extension (TCE) of it by the circle group T. 1 T E m G 1 Ideal situation: Given G, there is a group G (Univercal TCE) above G (G G) such that every PUR of G lifts to a UR of G, and G is a central extension of G. Already in 1939, Wigner proved that all PUR s of the Poincaré group P = R 4 SO(1, 3) 0 lift to UR s of the simply connected (2-fold) covering group P of the Poincaré group. In other words, P = R 4 SL(2, C) R is already the universal TCE of the Poincare group (UTCE). Jukka Virtanen (UCLA) Lie Algebras Workshop 10 / 43

Groups over non-archimedean fields Not all groups have UTCE s! If G is an algebraic group defined over a local field k, G(k) often does not have UTCEs. For a lcsc group to have a UTCE it is necessary that the commutator subgroup should be dense in it. Over a non-archimedean local field, the commutator subgroups of the Poincare group and the orthogonal groups are open and closed proper subgroups and so they do not have UTCE s. Also while may be surjective, need not be. G 1 ( k) G( k) G 1 (k) G(k) Jukka Virtanen (UCLA) Lie Algebras Workshop 11 / 43

Multipliers Let G be an lcsc group and T be the group of complex numbers of unit modulus. U(x)U(y) =m(x, y)u(xy) Definition A multiplier for G is a Borel map m : G G T such that: 1 m(x, yz)m(y, z)=m(xy, z)m(x, y) for all x, y G. 2 m(x, 1) =m(1, x) =1 for all x G. Jukka Virtanen (UCLA) Lie Algebras Workshop 12 / 43

Multipliers Multipliers of G form a commutative group, Z 2 (G) under pointwise multiplication. Multipliers m and m are called equivalent if there exists a Borel map a : G T (a(1) =1) such that m (x, y) =m(x, y) a(x)a(y) a(xy) If U is an m representation and U (x) =a(x)u(x), where a : G T, then U is an m representation and m and m are equivalent. If a multiplier m is equivalent to 1, then m(x, y) = a(xy) a(x)a(y), and we say that m is a trivial multiplier. The subgroup of trivial multipliers is denoted by B(G). We define the multiplier group of G as H 2 (G) =Z 2 (G)/B(G). Jukka Virtanen (UCLA) Lie Algebras Workshop 13 / 43

Multipliers for Semidirect Products We are mostly interested in semidirect product groups. H = A G where A and G are lcsc groups and A is abelian. We consider multipliers of H that are trivial when restricted to A. Let M A (H) be the group of multipliers of H such that for m M A (H), m A A = 1. Let H 2 A (H) denote its image in H2 (H). For any multiplier m of H with m A A = 1, [m A ] is G-invariant. In most cases, 1 is the only class in H 2 (A) which is G-invariant. Determine H 2 A (H). Determine m-representations for each m Z 2 (H). Jukka Virtanen (UCLA) Lie Algebras Workshop 14 / 43

Notation for 1-Cocycles Let A be the character group of A. Definition A 1-cocycle for G with coefficients in A is a continuous map f : G A such that f (gg )=f (g)+g[f (g )] (g, g G) Denote the abelian group of continuous 1-cocycles by Z 1 (G, A ). The coboundaries are the cocycles of the form g g[a] a for some a A. The coboundaries form a subgroup B 1 (G, A ) of Z 1 (G, A ). Form the cohomology group H 1 (G, A )=Z 1 (G, A )/B 1 (G, A ). Jukka Virtanen (UCLA) Lie Algebras Workshop 15 / 43

Description of the Multipliers of H The following theorem describes the multipliers of H. Theorem (Mackey) HA 2 (H) H2 (G) H 1 (G, A ) m (m 0,θ) m(ag, a g )=m 0 (g, g )θ(g 1 )(a ) Jukka Virtanen (UCLA) Lie Algebras Workshop 16 / 43

Mackey machine of regular semidirect products Set up: H = A G. A, G lcsc groups. A abelian. Theorem If the action of G on A is regular then: Orbits G[χ] χ A UIRs of H Stabilizer of χ in G is G χ UIRs of G χ Jukka Virtanen (UCLA) Lie Algebras Workshop 17 / 43

Affine action We consider a group H = A G where G and A are lcsc groups and A is abelian. Lemma (V.S.V, V. 2009) Let θ : G A be a continuous map with θ(1) =0. Define g θ {χ} = g[χ]+θ(g), for g G,χ A. Then a θ :(g,χ) g θ {χ} defines an action of G on A if and only if θ Z 1 (G, A ). Definition The action a θ :(g,χ) g θ {χ} is called the affine action of G on A determined by θ. Remark: Suppose H 1 (G, A )=0. Then the affine action reduces to the ordinary action. Jukka Virtanen (UCLA) Lie Algebras Workshop 18 / 43

Mackey machine for m-representations Set up as before: H = A G. A and G are lcsc groups over local fields with A abelian. Multipliers of H are trivial when restricted to A A. Theorem (V.S.V, V. 2009) If the action of G on A is regular then: Orbits G{χ} χ A Irreducible m-representations of H Stabilizer of χ in G is G χ Irreducible m 0 representations of G χ Jukka Virtanen (UCLA) Lie Algebras Workshop 19 / 43

Poincaré Group Let V be finite dimensional, isotropic quadratic vector space over a field k of ch 2. Let G = SO(V ) be the group of k-points of the corresponding orthogonal group preserving the quadratic form. By the k-poincaré group we shall mean the group P V = V G. It is the group of k-points of the corresponding algebraic group. From now on we assume that k is Q p. Jukka Virtanen (UCLA) Lie Algebras Workshop 20 / 43

Adapting the m-mackey machine theorem to the p-adic Poincaré group To use the m-mackey machine to describe the PUIRs of the Poincaré group we will establish the following: 1 Replace V by V. 2 The cohomology H 1 (SO(V ), V ) is trivial. 3 We need that the action of SO(V ) on V is regular. 4 The multipliers of P V must be trivial when restricted to V. 5 All of the orbits admit invariant measures. Jukka Virtanen (UCLA) Lie Algebras Workshop 21 / 43

Particle classification theorem for the Poincaré group Every multiplier for P V is the lift to P V of a multiplier for SO(V ), upto equivalence. (Recall: H 2 A (H) H2 (G) H 1 (G, A )) Theorem Orbits SO(V )[p] p V Stabilizer of p in SO(V ) is SO(V ) Irreducible m-representations of P V p Irreducible m p representations of SO(V ) p Jukka Virtanen (UCLA) Lie Algebras Workshop 22 / 43

Orbits Since the quadratic form on V is nondegenerate, V V. The action of SO(V ) on V goes over to the action of SO(V ) on V. Quadratic form on V is invariant under SO(V ), the level sets of the quadratic form are invariant sets. Under SO(V ), V decomposes into invariant sets of the following types. 1 The sets M a = {p V (p, p) =a 0}. (Massive orbits) 2 The set M 0 = {p V (p, p) =0, p 0}. (Massless orbit) 3 The set {0}. (Trivial massless) Lemma If dim(v ) 3, the sets M a,m 0 and {0} are all the orbits. Moreover, the action is regular. Jukka Virtanen (UCLA) Lie Algebras Workshop 23 / 43

Summmary of the the particle classification for the Poincaré group Multipliers Multipliers are those of SO(V ). Stabilizers For massive point p V we may write V = U < p > and the stabilizer is SO(V ) p = SO(U) For massless point SO(V ) p = P W where W is a quadratic vector space Witt equivalent to V and dim(w )=dim(v ) 2. Representations Representations of P V are obtained by finding the m-representations of the stabilizers. Jukka Virtanen (UCLA) Lie Algebras Workshop 24 / 43

Witt equivalence Let V 1 and V 2 be two quadratic vector spaces. If V 1 = A 1 H 1 V 2 = A 2 H 2 where A 1, A 2 are anisotropic, H 1, H 2 are maximum hyperbolic subspaces and A 1 A 2. Then V 1 and V 2 are Witt equivalent. Definition A quadratic vector space V of dimension 2n is called a hyperbolic space if there is an orthogonal basis (e i, f i ) 1 i n such that ei 2 = fi 2 = 0, (e i, f j )=δ ij for all i, j. Jukka Virtanen (UCLA) Lie Algebras Workshop 25 / 43

Massive and massless particles Definition A PUIR of the Poincaré group is called an elementary particle. A particle which corresponds to the orbit of a vector p V is called massless if p 0 and is massless ((p, p) =0), trivial if p = 0, and massive if p is massive ((p, p) 0). Jukka Virtanen (UCLA) Lie Algebras Workshop 26 / 43

Galilean group over Q p Analogue of the Galilean group over Q p V = V 0 V 1 is a finite-dimensional vector space over Q p. Where V 0 is an isotropic quadratic vector space and V 1 Q p. The Galilean group is G = V (V 0 SO(V 0 )). Technically pseudo-galilean group. Jukka Virtanen (UCLA) Lie Algebras Workshop 27 / 43

Particle classification of the p-adic Galilean group As in the case for the Poincaré group one can show that the criteria for the m-mackey machine are satisfied. H 1 (V 0 SO(V 0 ), V ) Q p. H 2 (G) H 2 (SO(V 0 )) Q p Affine action comes into play! In our paper we find the multipliers orbits and the stabilizers. The representations are parameterized by by τ( 0) Q p and the projective representations μ of SO(V 0 ). We interpret τ as the Schrödinger mass and μ as the spin. When τ = 0 there is a close resemblance to the real case where the representations are ruled unphysical. Jukka Virtanen (UCLA) Lie Algebras Workshop 28 / 43

Poincaré and Conformal groups Over R One can compactify spacetime: Imbed R m,n as a smooth variety in P(R m+1,n+1 ). SO(m + 1; n + 1) acts conformally and transitively on the compactified spacetime. The conformal group is important because it is the symmetry group for radiation. Maxwell s equations are invariant under the conformal group. P = R m,n SO(m, n) SO(m + 1, n + 1) Over Q p P = V SO(V ) SO(W ) Here dim(w )=dim(v )+2 and W and V are Witt equivalent. Jukka Virtanen (UCLA) Lie Algebras Workshop 29 / 43

Conformal Symmetry Definition If a PUIR U of P V extends to a PUIR U of the conformal group we say that U has conformal symmetry. Jukka Virtanen (UCLA) Lie Algebras Workshop 30 / 43

Conformal symmetry over R Problem solved completely over R. E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, and independently many others for Minkowski 4-space. Angelopoulous and Laoues for Minkowski n-space. Only massless particles have conformal symmetry, massive ones do not. Remark: Mass is invariant quantity under the Poincaré group, but conformal group can dilate mass. Jukka Virtanen (UCLA) Lie Algebras Workshop 31 / 43

Impossibility of Conformal symmetry for Massive Particles over Non-Archimedean spacetime Theorem (V. and V. 2009) A massive PUIR of p-adic Poincaré group does not have conformal symmetry It is an open question as to whether or not massless particles of the p-adic Poincaré group have conformal symmetry. Jukka Virtanen (UCLA) Lie Algebras Workshop 32 / 43

Thank you! My thanks to you! Jukka Virtanen (UCLA) Lie Algebras Workshop 33 / 43

m-systems of imprimitivity Let G be a lcsc group. Let X be a G-space that is also a standard Borel space. Let H be a separable Hilbert space. Definition An m-system of imprimitivity is a pair (U, P), where P(E P E ) is a projection valued measure (pvm) on the class of Borel subsets of X, the projections being defined in H, and U is an m-representation of G in H such that U(g)P(E)U(g) 1 = P(g[E]) g G and all Borel E X. The pair (U, P) is said to be based on X. Jukka Virtanen (UCLA) Lie Algebras Workshop 34 / 43

m-systems of imprimitivity and m-representations Let X to be a transitive G-space. We fix some x 0 X and let G 0 be the stabilizer of x 0 in G, so that X G/G 0. We will also fix a multiplier m for G and let m 0 = m G0 G 0. Theorem There is a natural one to one correspondence between m 0 -representations μ of G 0 and m-systems of imprimitivity S μ := (U, P) of G based on X. Under this correspondence, we have a ring isomorphism of the commuting ring of μ with that of S μ, so that irreducible μ correspond to irreducible S μ. Jukka Virtanen (UCLA) Lie Algebras Workshop 35 / 43

Mackey machine for m-systems of imprimitivity Theorem Fix θ Z 1 (G, A ) and m M A (H), m (m 0,θ). Then there is a natural bijection between m-representations V of H = A G and m 0 -systems of imprimitivity (U, P) on A for the affine action g,χ g θ {χ}, defined by θ. The bijection is given by: V (ag) =U(a)U(g), U(a) = a,χ dp(χ). A Jukka Virtanen (UCLA) Lie Algebras Workshop 36 / 43

Mackey machine for m-representations Theorem Let X = G{χ 0 } and λ be a σ-finite quasi-invariant measure for the action of G. Then, for any irreducible m 0 -representation μ of G χ0 in the Hilbert space K, the corresponding m-representation V acts on L 2 (X, K,λ) and has the following form: (V (ag)f )(χ) = a,χ ρ g (g 1 {χ}) 1 2 )δ(g, g 1 {χ})f (g 1 {χ}) where δ is any strict m 0 -cocycle for (G, X) with values in U, the unitary group of K, such that δ(g,χ 0 )=μ(g), g G χ0. We note that the ρ g -factors drop out if λ is an invariant measure. Jukka Virtanen (UCLA) Lie Algebras Workshop 37 / 43

Replace V by V We shall replace V by the algebraic dual V of V. Since V, the topological dual of V, is isomorphic to the algebraic dual V, the isomorphism being natural and compatible with actions of GL(V ). The isomorphism is easy to set up but depends on the choice of a non-trivial additive character on Q p,sayψ. Once we fix ψ, then, for any p V, χ p : a ψ( a, p ) is in V, and p χ p is a topological group isomorphism of V with V. Jukka Virtanen (UCLA) Lie Algebras Workshop 38 / 43

The cohomology H 1 (SO(V ), V ) is trivial Proof consists of showing that H 1 (so(v ), V )=0 due to the fact that so(v ) is semi simple. Then it is possible to show that the map : H 1 (SO(V ), V ) H 1 (so(v ), V ) is injective. So we have H 1 (SO(V ), V )=0 and so H 2 (H) =H 2 (G). Every multiplier of P V is equivalent to the lift of a multiplier of SO(V ). Jukka Virtanen (UCLA) Lie Algebras Workshop 39 / 43

Invariant measure Lemma For V of any dimension 1, all the orbits of SO(V ) admit invariant measures. Sketch of proof: If G be a unimodular lcsc group, and H is a closed subgroup of G; then for G/H to admit a G-invariant measure unimodularity of H is a sufficient condition. Let p V, let L p be its stabilizer Case 1) Massive p: Let p M a. Write V = U < p >. Then L p SO(U). SO(U) is semisimple hence unimodular. Case 2) Massless p: Let p M 0. It turns out that L p P W where P W is the Poincaré group of a quadratic vector space W (with dim(w )=dim(v ) 2 and W is Witt equivalent to V ). P W is unimodular. Jukka Virtanen (UCLA) Lie Algebras Workshop 40 / 43

The multipliers of P V are trivial when restricted to V Our theorem requires that the multipliers of P V be trivial when restricted to V. Sketch of proof: Let H = A G. Λ 2 (A) are the alternating bicharacters of A. If H is 2-regular then Λ 2 (A) H 2 (A) If 1 is the only element of Λ 2 (A) invariant under G then all multipliers of H restrict to A as trivial. Alternating bicharacters are in one to one correspondence with skew symmetric bilinear forms. For any skew symmetric bilinear form b on V V, ψ(b) is a multiplier for V, and the map b ψ(b) induces an isomorphism of Λ 2 (V ) with H 2 (V ). V is irreducible under SO(V ) and admits a symmetric invariant bilinear form, namely (, ). Any invariant bilinear form must be a multiple of this, and so, any skew symmetric invariant bilinear form must be 0. Jukka Virtanen (UCLA) Lie Algebras Workshop 41 / 43

Example Let G = SL(2, Q p ). The adjoint representation exhibits G as the spin group corresponding to the quadratic vector space g which is the Lie algebra of G equipped with the Killing form. The adjoint map G G 1 = SO(g) is the spin covering for SO(g) but this is not surjective; in the standard basis X = ( 0 1 0 0 ), H = ( 1 0 0 1 ), Y = ( 0 0 1 0 ) Jukka Virtanen (UCLA) Lie Algebras Workshop 42 / 43

Example the spin covering map is ( a b c d 2 ) a2 ac 2ab ad + bc b bd c 2 2cd d 2. The matrix α 0 0 0 1 0 0 0 α 1 ( a b is in SO(g); if it is the image of c d α = a 2, so that unless α Q p ), then b = c = 0, d = a 1, and 2, this will not happen. Jukka Virtanen (UCLA) Lie Algebras Workshop 43 / 43