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The Role of Orthogonal and Symplectic Clifford

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The Role of Orthogonal and Symplectic Clifford THE ROLE OF ORTHOGONAL AND SYMPLECTIC CLIFFORD ALGEBRAS IN QUANTUM FIELD THEORY Matej Pavšič J. Stefan Institute, Ljubljana, Slovenia Contents - Introduction - Spaces with orthogonal and symplectic forms - Heisenberg equations as equation of motion for basis vectors - Point particle in `superspace’ - Description of fields Orthogonal case Symplectic case - Discussion: Prospects for quantum gravity - Conclusion Introduction Persisting problem of quantum gravity and the unification of interactions. A need to reformulate the conceptual foundations of physics and to employ a more evolved mathematical formalism. In this talk I will consider Clifford algebras which provide very promising tools for description and generalization of geometry and physics. Orthogonal Clifford algebra The inner, symmetric , product of basis vectors 1 γ a gives the orthogonal metric, . γ a ·γb≡2 ( γγ ab + γγ ba ) = g ab gab 1 The outer, antisymmetric , product of basis vectors γa∧γb ≡2 ( γγ ab − γ b γ a ) gives the basis bivector. Symplectic Clifford algebra The inner, antisymmetric , product of basis vectors q 1 a gives the symplectic metric, J ab . qab∧q ≡2 ( q ab qqq− b a ) = J ab 1 The outer, symmetric , product of basis vectors qab·q ≡ 2 (q ab qqq+ b a ) gives a symplectic bivector. The generators of an orthogonal Clifford algebra can be transformed into a basis in which they behave as fermionic creation and annihilation operators. The generators of a symplectic Clifford algebra behave as bosonic creation and annihilation operators. We will show how both kinds of operators can be united into a single structure so that they form a basis of a `superspace’. We will consider an action for a point particle in such superspace. Instead of finite dimensional spaces, we can consider infinite dimensional spaces. Then we have description of a field theory in terms of fermionic and bosonic creation and annihilation operators. The latter operators can be considered as being related to the basis vectors of the corresponding infinite dimensional space. Spaces with orthogonal an symplectic forms I. Orthogonal case (,)ab= ( ababγγ , ) = a a (,) γγ bagb bab = µ g a bg abg ab a= a γ µ (γa , γ b ) g= g ab metric For a basis we can take generators of the Basis vectors orthogonal Clifford algebra: 1 (,)γ γ=g = ( γ γ + γ γ ) = γ· γ = g abg ab2 ab ba a b ab Vectors are Clifford numbers: 1 (,)a b= ( ab + ba ) = a ·b g 2 Inner product of vectors a and b II. Symplectic case ab a bab (,')(zzJ = zqza ,')q bJ = zqqz (,)' a b J = zJzab ' a z= z q a symplectic metric (qa , q b ) J = J ab Symplectic basis vectors For a symplectic basis we can take generators of the symplectic Clifford algebra: 1 (,)qq= J = ( qq −q q ) = q ∧ q = J a b J ab2 ab b a a b a b Vectors are now symplectic Clifford numbers: 1 (,')zz= ( zzzz '− ') = z ∧ z ' J 2 Inner product of symplectic vectors z and z’ Explicit notation with coordinates and momenta za = ( xµ , p µ ) a µ()x µ () p Symplectic vector z= zqa = xqµ + pq µ a b a 1 b (,')zzJ= zqq (,)ab J z ' = z 2 (q abbaq− qq) z ' a b = z Jab z ' 0 ηµν µ νν ν J = =(xp ' − px ') ηµν ab −ηµν 0 1 Relations give 2 [qa , q b ] = J ab 1()()xx 1 ()( pp ) 2[qqµν , ]0,= 2 [ qq µν , ]0 = 1 (x )( p ) Heisenberg commutation 2 [qµ , q ν ] =η µν relations Poisson bracket (symplectic case) ∂f ∂ g {f , g } ≡ J ab PB ∂za ∂ z b By introducing the symplectic basis vectors, we can rewrite the above expression as Symplectic metric 1 ∂fa ∂ g b ∂fab ∂ g [a q, b q ] = aJ b 0 −ηµν 2 ∂zz∂ ∂ z ∂ z J ab = ηµν 0 c d If we take fz=, gz = 0 ηµν 1 c d cd J ab = [q , q ] = J −ηµν 0 2 These are the Heisenberg commutation c d relations for `operators’ q and q . a a a q are thus `quantized’ phase space coordinates z . z are real coordinates Poisson bracket (orthogonal case) ∂f ∂ g {f , g } ≡ g ab PB ∂λa ∂ λ b By introducing the basis vectors we can rewrite the above expression as Orthogonal metric 1 ∂fa ∂ g b ∂f a b ∂ g γ, γ = g η µν 0 2 ∂λa∂ λ b ∂λ a ∂ λ b g ab = µν 0 η c d If we take f=λ, g = λ ηµν 0 1cd 1 cd dc cd gab = {γ ,γ} ≡ ( γγ+ γγ ) = g 0 ηµν 2 2 These are the anticommutation relations for c d `operators’ γ and γ . a a a γ are thus `quantized’ λ . λ are real anticommuting coordinates Representation of operators I. Orthogonal Clifford algebra η 0 1 µν gab = γγab·≡ ( γγ ab + γγ ba ) = g ab 2 0 ηµν In even dimensions we can write: γa =( γγµ , µ ), µ = 0,1,2,3 We can introduce Witt basis: 1 θ= ( γ+ i γ ) µ2 µ µ 1 θ= ( γ− i γ ) µ2 µ µ 1 θθ·= ( θθθθ+= ),·0,·0 η θθ = θθ = µν2 µννµ µν µν µν γ µ,γ µ , θµ, θ µ can be represented: 1) as 4 x 4 matrices, 2) in terms of Grassmann coordinates: µ µν µ µ ∂ θ→2 ξ , θ → 2 θ= η θ ν µ ∂ξ µ ξξµν+ ξξ νµ = 0 II. Symplectic Clifford algebra 1 qa∧≡ q b( qqqq ab − ba ) = J ab 0 ηµν 2 J ab = −ηµν 0 We can write: ()x () p qa =( qµ , q µ ),µ = 0,1,2,3 1 q()x∧ q () p= (q ()() xp q − q ()() pxq ) =η µν2 µ ν µν µν ()xx () () pp () qµ ∧qν = 0,q µ∧ q ν = 0 Then the operators cannot ()x () p can be represented: be cast into Hermitian form qµ, q µ 1) as 4 x 4 matrices, 2) in terms of commuting coordinates: ∂ µ()x µ () p µ( x) µν (x ) q →2x , q µ → 2 µ q=η q ∂x ν xxµν− xx νµ = 0 Heisenberg equations as equations of motion for basis vectors Let us now consider the action za = ( xµ , p µ ) I=1 dzJzτ (ɺa b + zKz a b ) H= 1 zKza b 2 ∫ ab ab 2 ab δ za ∂H zɺa= J ab ∂zb a Let us consider trajectories z ( τ ) as components of an infinite dimensional vector: qa()τ∧ q b (') τ = J ab ()(') ττ zzq= a(τ ) ≡ dzτ a () τ q () τ a(τ ) ∫ a =J ab δ( τ − τ ′ ) ɺa()τ a () τ ɺ zqa()τ= − zq a () τ and write the action in the form 1a()τ ba (') ττ 1 () b (') τ H 1 a b ɺ = 2 dzτ() τ Kzab () τ IzJ=2ab()(')ττ z + 2 zK ab ()(') ττ z ∫ 1 a(τ ) b ( τ ') = z Ka(τ ) b ( τ ') z The equations of motion are: 2 K= K δ( τ − τ ′ ) ∂H ab(τ ) ( τ ') ab zqɺa()τ=− zqqJ a () τɺ = ab ()(') ττ =− qKz a () τ b (') τ a()()()τ a τ a τ ∂zb(τ ') a(τ ) b ( τ ') a(τ ) a b(τ ')ɺ a ( τ ) b ( τ ') q≡ q (τ ) zqb(τ ')= qK a ( τ ) b ( τ ') z b(τ ) This equation holds for any z ɺ a(τ ) qb(τ ')= q K a ( τ ) b ( τ ') Equations of motion for operators qa(τ ) Kab(τ ) ( τ ') = K ab δ( τ − τ ′ ) ɺ a Kab= K ba qb()τ= q () τ K ab ˆ 1 a b H= 2 qKqab Symplectic inner product ˆ b [qHa , ] = Kq ab is given by commutator ɺ ˆ qa= [ q a , H ] Heisenberg equations of motion We see that the basis vectors of phase space satisfy the Heisenberg equations of motion for `quantum’ operators qa(τ ) ≡ q a (τ ) b(τ ') a ( τ ) b ( τ ') ɺ a(τ ) a zqb(τ ')= qK a ( τ ) b ( τ ') z z≡ z (τ ) b(τ ) This equation hold for any z ɺ a(τ ) qb(τ ')= q K a ( τ ) b ( τ ') Equations of motion for operators qa(τ ) K= K δ( τ − τ ′ ) ab(τ ) ( τ ') ab a(τ ) That the operator equations hold for any z means ɺ a that coordinates and momenta are undetermined.Kab= K ba qb()τ= q () τ K ab ˆ 1 a b H= 2 qKqab b Symplectic inner product [qH ,ˆ ] = Kqa µ µ az ab()τ= ( x (), τ p ())is τ given by commutator ɺ ˆ qa= [ q a , H ] Heisenberg equations of motion We see that the basis vectors of phase space satisfy the Heisenberg equationsClassical of motion solution for Any trajectory `quantum’ operators Point particle action in `superspace’ We introduce the generalized vector space whose elements are: A a a a µ µ a µ µ A z= ( z ,λ ), z= ( x , x ) , λ= ( λ , λ ) Z= z q A qA= (qa ,γ a ), qa = (qµ , q µ ), γa = ( γ µ ,γ µ ) coordinates basis elements symplectic orthogonal part J ab 0 part 〈qAq B 〉= S G AB = 0 gab 0 ηµν qa ∧ qb = J ab = symplectic metric −ηµν 0 ηµν 0 γ a ·γ b =gab = orthogonal metric 0 ηµν Let us consider a particle moving in such space. Its worldline is given by: zA= Z A (τ ) parameter on the worldline ) Example of a possible action Example of a possible action za = ( zµ , z µ ) 1A B 1 A B J abɺ0 A a. ɺ a I= d τ〈ZqZAqB 〉 S = d τ ZGZ AB 2∫GAB = , 2 ∫ z = (,) z λ a µ µ a bɺ a0 ɺ g a λ= ( λ , λ ) I= dτ ( zJzɺ ɺ + λ g λ )ab ∫ ab ab J 0 µ = 0,1,2,3 G=ab , zA = (,) z a.
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