structure of elementary particles in non-archimedean...

Structure of Elementary Particles inNon-Archimedean Spacetime

Jukka Virtanen

Department of MathematicsUniversity of California Los Angeles

USC Lie Groups, Lie Algebras and their Representations WorkshopLos 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, andConformal Symmetry, of Elementary Particles over Non-ArchimedeanSpaceTime, Letters in Mathematical Physics, (2009), 171 - 182.

V. S. Varadarajan and J. Virtanen, Structure, Classification, andConformal Symmetry, of Elementary Particles over Non-ArchimedeanSpaceTime, p-adic numbers, ultrametric analysis and applications Vol. 2No. 2 2010

V. S. Varadarajan and J. Virtanen, Structure, classification, and conformalsymmetry of elementary particles over non-archimedean space-time,arXiv:1002.0047v1 [math-ph], 2010.

V. S. Varadarajan, Multipliers for the symmetry groups of p-adicspace-time, p-Adic Numbers, Ultrametric Analysis, and Applications,1(2009), 69–78.


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 Poincare group.8 Elementary particles of the p-adic Galilean group.9 Some remarks on conformal symmetry of p-adic particles.


Some prophetic remarks

Now it seems that the empirical notions on which the metric determinations ofspace are based, the concept of a solid body and light ray, lose their validity inthe infinitely small; it is therefore quite definitely conceivable that the metricrelations of space in the infinitely small do not conform to the hypothesis ofgeometry; and in fact, one ought to assume this as soon as it permits asimpler way of explaining phenomena. -Bernhard Riemann, Inaugurallecture, 1854

In fact, I would not be too surprised if discrete mod p mathematics and thep-adic numbers would eventually be of use in the building of models for verysmall phenomena. -Raoul Bott, 1975


Motivation

Bit of historical introduction

Non-Archimedian geometry developed in 19th century to answerquestions 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 theunderlying structures are over R or C over the new fields.

Arithmetic physics is the study of well known algebraic structures inquantum mechanics over new fields and rings.

In 1930s Weyl considered quantum mechanics over finite fields.

In 1970s Beltrametti and his collaborators started to investigatealternative possibilities for the micro-structure of spacetime based onp-adic fields.


Volovich hypothesis

How small is small?

Planck length (10−33cm) and Planck time (10−44s).

Planck length is the smallest possible measurable distance and Plancktime the smallest measurable time.

Planck length arises from interplay between general theory of relativityand quantum mechanics

To probe region of space on sub-Planck scale one needs energy greaterthan Planck mass. This will create a ”miniature black hole”.

Volovich (1987)

Archimedean axiom is at its core a statement about comparison oflengths.

At sub-Planck scale the Archimedean property breaks down.

At Planck scale spacetime itself fluctuates and transitions from R to Qp

cannot be ruled out.

New geometry of spacetime based on a non-Archimedean field.


Main theme of the Talk

We want to describe the elementary particles under the assumption that theunderlying field is a non-Archimedean field.


Elementary Particles, Symmetry Groups andProjective 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:

p([ψ], [φ]) = |(ψ, φ)|2, ψ, φ ∈ H.

Symmetries of a quantum system S.

DefinitionA symmetry of a quantum system is a bijection P(H) → P(H) that preservesp([ψ], [φ]) for any two [ψ], [φ] ∈ P(H).


Projective Unitary Representations (PUR)

Symmetries are induced by unitary or anti-unitary operators, unique up tophase.

R : H → H, [φ] → [Rφ]

As a consequence that the symmetry of a quantum system with respectto a group G may be expressed by a projective unitary representation(PUR) of G.

DefinitionA projective unitary representation of G is a map U of G to the unitary groupU of H such that U(1) = 1 and U(x)U(y) = m(x , y)U(xy). m is Borel mapfrom 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 projectiveunitary representations (PUIRs).


Topological Central Extensions

A PUR may be lifted to an ordinary unitary representation (UR) of asuitable topological central extension (TCE) of it by the circle group T .

1 → T → Em → 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 acentral extension of G.

Already in 1939, Wigner proved that all PUR’s of the Poincare groupP = R4 � SO(1, 3)0 lift to UR’s of the simply connected (2-fold) covering groupP∗ of the Poincare group. In other words, P∗ = R4 � SL(2,C)R is already theuniversal TCE of the Poincare group (UTCE).


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 nothave UTCEs.

For a lcsc group to have a UTCE it is necessary that the commutatorsubgroup should be dense in it.

Over a non-archimedean local field, the commutator subgroups of thePoincare group and the orthogonal groups are open and closed propersubgroups and so they do not have UTCE’s.

Also while

G1(k) → G(k)

may be surjective,

G1(k) → G(k)

need not be.


Multipliers

Let G be an lcsc group and T be the group of complex numbers of unitmodulus.

U(x)U(y) = m(x , y)U(xy)

DefinitionA 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.


Multipliers

Multipliers of G form a commutative group, Z2(G) under pointwisemultiplication.

Multipliers m and m′ are called equivalent if there exists a Borel mapa : 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 , thenU ′ 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 themultiplier group of G as H2(G) = Z 2(G)/B(G).


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 MA(H) be the group of multipliers of H such that for m ∈ MA(H),m|A×A = 1. Let H2

A(H) denote its image in H2(H).

For any multiplier m of H with m|A×A = 1, [mA] is G-invariant. In mostcases, 1 is the only class in H2(A) which is G-invariant.

Determine H2A(H).

Determine m-representations for each m ∈ Z 2(H).


Notation for 1-Cocycles

Let A∗ be the character group of A.

DefinitionA 1-cocycle for G with coefficients in A∗ is a continuous map f : G → A∗ suchthat

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 somea ∈ A∗.

The coboundaries form a subgroup B1(G,A∗) of Z 1(G,A∗).Form the cohomology group H1(G,A∗) = Z 1(G,A∗)/B1(G,A∗).


Description of the Multipliers of H

The following theorem describes the multipliers of H.

Theorem(Mackey)

H2A(H) � H2(G) × H1(G,A∗)

m ↔ (m0, θ)

m(ag, a′g′) = m0(g, g′)θ(g−1)(a′)


Mackey machine of regular semidirect products

Set up:

H = A � G.

A, G lcsc groups. A abelian.

TheoremIf the action of G on A∗ is regular then:

UIRs of H ⇔⎧⎨⎩

Orbits G[χ] χ ∈ A∗

Stabilizer of χ in G is Gχ

UIRs of Gχ


Affine action

We consider a group H = A � G where G and A are lcsc groups and A isabelian.

Lemma(V.S.V, V. 2009) Let θ : G → A∗ be a continuous map with θ(1) = 0. Definegθ{χ} = g[χ] + θ(g), for g ∈ G, χ ∈ A∗. Then aθ : (g, χ) �→ gθ{χ} defines anaction of G on A∗ if and only if θ ∈ Z1(G,A∗).

DefinitionThe action aθ : (g, χ) �→ gθ{χ} is called the affine action of G on A∗

determined by θ.

Remark: Suppose H1(G,A∗) = 0. Then the affine action reduces to theordinary action.


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:

Irreducible m-representations of H ⇔⎧⎨⎩

Orbits G{χ} χ ∈ A∗

Stabilizer of χ in G is Gχ

Irreducible m0 representations of Gχ


Poincare Group

Let V be finite dimensional, isotropic quadratic vector space over a field k ofch = 2. Let G = SO(V ) be the group of k-points of the correspondingorthogonal group preserving the quadratic form. By the k-Poincare group weshall mean the group

PV = V � G.

It is the group of k-points of the corresponding algebraic group.

From now on we assume that k is Qp.


Adapting the m-Mackey machine theorem to thep-adic Poincare group

To use the m-Mackey machine to describe the PUIRs of the Poincare groupwe will establish the following:

1 Replace V∗ by V ′.2 The cohomology H1(SO(V ),V ′) is trivial.3 We need that the action of SO(V ) on V ′ is regular.4 The multipliers of PV must be trivial when restricted to V .5 All of the orbits admit invariant measures.


Particle classification theorem for the Poincare group

Every multiplier for PV is the lift to PV of a multiplier for SO(V ), up toequivalence. (Recall: H2

A(H) � H2(G) × H1(G,A∗))

Theorem

Irreducible m-representations of PV ⇔

⎧⎪⎪⎨⎪⎪⎩

Orbits SO(V )[p] p ∈ V ′

Stabilizer of p in SO(V ) is SO(V )p

Irreducible mp representations ofSO(V )p


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 thequadratic form are invariant sets.

Under SO(V ′), V ′ decomposes into invariant sets of the following types.

1 The sets Ma = {p ∈ V ′ | (p, p) = a = 0}. (Massive orbits)2 The set M0 = {p ∈ V ′ | (p, p) = 0, p = 0}. (Massless orbit)3 The set {0}. (Trivial massless)

Lemma

If dim(V ) ≥ 3, the sets Ma, M0 and {0} are all the orbits. Moreover, the actionis regular.


Summmary of the the particle classification for thePoincare group

Multipliers

Multipliers are those of SO(V ).

Stabilizers

For massive point p ∈ V ′ we may write V = U⊕ < p > and the stabilizeris SO(V )p = SO(U)

For massless point SO(V )p = PW where W is a quadratic vector spaceWitt equivalent to V and dim(W ) =dim(V ) − 2.

Representations

Representations of PV are obtained by finding the m-representations of thestabilizers.


Witt equivalence

Let V1 and V2 be two quadratic vector spaces. If

V1 = A1 ⊕ H1

V2 = A2 ⊕ H2

where A1,A2 are anisotropic, H1, H2 are maximum hyperbolic subspaces andA1 � A2. Then V1 and V2 are Witt equivalent.

DefinitionA quadratic vector space V of dimension 2n is called a hyperbolic space ifthere is an orthogonal basis (ei , fi)1≤i≤n such that e2

i = f 2i = 0, (ei , fj) = δij for

all i , j .


Massive and massless particles

DefinitionA PUIR of the Poincare group is called an elementary particle. A particlewhich corresponds to the orbit of a vector p ∈ V ′ is called massless if p = 0and is massless ((p, p) = 0), trivial if p = 0, and massive if p is massive((p, p) = 0).


Galilean group over Qp

Analogue of the Galilean group over Qp

V = V0 ⊕ V1 is a finite-dimensional vector space over Qp. Where V0 is anisotropic quadratic vector space and V1 � Qp.

The Galilean group is G = V � (V0 � SO(V0)).

Technically pseudo-Galilean group.


Particle classification of the p-adic Galilean group

As in the case for the Poincare group one can show that the criteria forthe m-Mackey machine are satisfied.

H1(V0 � SO(V0),V ′) � Qp.

H2(G) � H2(SO(V0)) × Qp

Affine action comes into play!

In our paper we find the multipliers orbits and the stabilizers.

The representations are parameterized by by τ(= 0) ∈ Qp and theprojective representations μ of SO(V0). We interpret τ as theSchrodinger mass and μ as the spin.

When τ = 0 there is a close resemblance to the real case where therepresentations are ruled unphysical.


Poincare and Conformal groups

Over R

One can compactify spacetime: Imbed Rm,n as a smooth variety inP(Rm+1,n+1).

SO(m + 1; n + 1) acts conformally and transitively on the compactifiedspacetime.

The conformal group is important because it is the symmetry group forradiation.

Maxwell’s equations are invariant under the conformal group.

P = Rm,n � SO(m, n) ↪→ SO(m + 1, n + 1)

Over Qp

P = V � SO(V ) ↪→ SO(W )

Here dim(W ) = dim(V ) + 2 and W and V are Witt equivalent.


Conformal Symmetry

DefinitionIf a PUIR U of PV extends to a PUIR U′ of the conformal group we say that Uhas conformal symmetry.


Conformal symmetry over R

Problem solved completely over R.

E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, andindependently 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 Poincare group, butconformal group can dilate mass.


Impossibility of Conformal symmetry for MassiveParticles over Non-Archimedean spacetime

Theorem(V. and V. 2009) A massive PUIR of p-adic Poincare group does not haveconformal symmetry

It is an open question as to whether or not massless particles of the p-adicPoincare group have conformal symmetry.


Thank you!

My thanks to you!


m-systems of imprimitivity

Let G be a lcsc group. Let X be a G-space that is also a standard Borelspace. Let H be a separable Hilbert space.

DefinitionAn m-system of imprimitivity is a pair (U,P), where P(E → PE ) is a projectionvalued measure (pvm) on the class of Borel subsets of X , the projectionsbeing 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 .


m-systems of imprimitivity and m-representations

Let X to be a transitive G-space. We fix some x0 ∈ X and let G0 be thestabilizer of x0 in G, so that X � G/G0. We will also fix a multiplier m for Gand let m0 = m|G0×G0 .

Theorem

There is a natural one to one correspondence between m0-representations μof G0 and m-systems of imprimitivity Sμ := (U,P) of G based on X. Underthis correspondence, we have a ring isomorphism of the commuting ring of μwith that of Sμ, so that irreducible μ correspond to irreducible Sμ.


Mackey machine for m-systems of imprimitivity

Theorem

Fix θ ∈ Z 1(G,A∗) and m ∈ M ′A(H), m � (m0, θ). Then there is a natural

bijection between m-representations V of H = A � G and m0-systems ofimprimitivity (U,P) on A∗ for the affine action g, χ �→ gθ{χ}, defined by θ. Thebijection is given by:

V (ag) = U(a)U(g), U(a) =

∫A∗〈a, χ〉dP(χ).


Mackey machine for m-representations

TheoremLet X = G{χ0} and λ be a σ-finite quasi-invariant measure for the action of G.Then, for any irreducible m0-representation μ of Gχ0 in the Hilbert space K,the corresponding m-representation V acts on L2(X ,K, λ) and has thefollowing form:

(V (ag)f )(χ) = 〈a, χ〉ρg(g−1{χ}) 12 )δ(g, g−1{χ})f (g−1{χ})

where δ is any strict m0-cocycle for (G,X ) with values in U , the unitary groupof K, such that δ(g, χ0) = μ(g), g ∈ Gχ0 .

We note that the ρg-factors drop out if λ is an invariant measure.


Replace V ∗ by V ′

We shall replace V∗ by the algebraic dual V ′ of V . Since V∗, the topologicaldual of V , is isomorphic to the algebraic dual V ′, the isomorphism beingnatural and compatible with actions of GL(V ). The isomorphism is easy to setup but depends on the choice of a non-trivial additive character on Qp, say ψ.Once we fix ψ, then, for any p ∈ V ′, χp : a �−→ ψ(〈a, p〉) is in V∗, and p �−→ χp

is a topological group isomorphism of V ′ with V∗.


The cohomology H1(SO(V ), V ′) is trivial

Proof consists of showing that H1(so(V ),V ′) = 0 due to the fact that so(V ) issemi simple. Then it is possible to show that the map∂ : H1(SO(V ),V ′) → H1(so(V ),V ′) is injective.

So we have H1(SO(V ),V ′) = 0 and so H2(H) = H2(G).

Every multiplier of PV is equivalent to the lift of a multiplier of SO(V ).


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 forG/H to admit a G-invariant measure unimodularity of H is a sufficientcondition. Let p ∈ V , let Lp be its stabilizerCase 1) Massive p:

Let p ∈ Ma. Write V = U⊕ < p >. Then Lp � SO(U).

SO(U) is semisimple hence unimodular.

Case 2) Massless p:

Let p ∈ M0. It turns out that Lp � PW where PW is the Poincare group of aquadratic vector space W (with dim(W ) = dim(V ) − 2 and W is Wittequivalent to V ).

PW is unimodular.


The multipliers of PV are trivial when restricted to V

Our theorem requires that the multipliers of PV be trivial when restricted to V .Sketch of proof:

Let H = A � G. Λ2(A) are the alternating bicharacters of A.

If H is 2-regular then Λ2(A) � H2(A)

If 1 is the only element of Λ2(A) invariant under G then all multipliers of Hrestrict to A as trivial.

Alternating bicharacters are in one to one correspondence with skewsymmetric bilinear forms.

For any skew symmetric bilinear form b on V × V , ψ(b) is a multiplier forV , and the map b → ψ(b) induces an isomorphism of Λ2(V ) with H2(V ).

V is irreducible under SO(V ) and admits a symmetric invariant bilinearform, namely (·, ·). Any invariant bilinear form must be a multiple of this,and so, any skew symmetric invariant bilinear form must be 0.


Example

Let G = SL(2,Qp). The adjoint representation exhibits G as the spin groupcorresponding to the quadratic vector space g which is the Lie algebra of Gequipped with the Killing form. The adjoint map G −→ G1 = SO(g) is the spincovering for SO(g) but this is not surjective; in the standard basis

X =

(0 10 0

), H =

(1 00 −1

), Y =

(0 01 0

)


Example

the spin covering map is

(a bc d

)�−→

⎛⎝ a2 −2ab −b2

−ac ad + bc bd−c2 2cd d2

⎞⎠ .

The matrix ⎛⎝ α 0 0

0 1 00 0 α−1

⎞⎠

is in SO(g); if it is the image of(

a bc d

), then b = c = 0, d = a−1, and

α = a2, so that unless α ∈ Q×p

2, this will not happen.


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