a theory of quantized fields based on orthogonal and symplectic clifford algebras

Adv. Appl. Cliff ord Algebras 22 (2012), 449–481© 2011 Springer Basel AG 0188-7009/020449-33published online September 15, 2011 DOI 10.1007/s00006-011-0314-4

A Theory of Quantized Fields Based onOrthogonal and Symplectic CliffordAlgebras

Matej Pavsic

Abstract. The transition from a classical to quantum theory is investi-gated within the context of orthogonal and symplectic Clifford algebras,first for particles, and then for fields. It is shown that the generators ofClifford algebras have the role of quantum mechanical operators thatsatisfy the Heisenberg equations of motion. For quadratic Hamiltonians,the latter equations are obtained from the classical equations of motion,rewritten in terms of the phase space coordinates and the correspond-ing basis vectors. Then, assuming that such equations hold for arbitrarypath, i.e., that coordinates and momenta are undetermined, we arrive atthe equations that contains basis vectors and their time derivatives only.According to this approach, quantization of a classical theory, formu-lated in phase space, is replacement of the phase space variables with thecorresponding basis vectors (operators). The basis vectors, transformedinto the Witt basis, satisfy the bosonic or fermionic (anti)commutationrelations, and can create spinor states of all minimal left ideals of thecorresponding Clifford algebra. We consider some specific actions forpoint particles and fields, formulated in terms of commuting and/oranticommuting phase space variables, together with the correspondingsymplectic or orthogonal basis vectors. Finally we discuss why such ap-proach could be useful for grand unification and quantum gravity.

Keywords. Orthogonal and symplectic Clifford algebras, phase space,quantization, quantum fields, unification of interactions.

1. Introduction

During last few decades we are faced with the persisting problems of quantumgravity and the unification of fundamental interactions. The situation revealsthe need to reformulate the conceptual foundations of physics and to employa more evolved mathematical formalism. It has turned out that Clifford alge-bras provide very promising tools for description [1]–[12] and generalization

Advances inApplied Cliff ord Algebras

450 M. Pavšic Adv. Appl. Clifford Algebras

of geometry and physics [13, 14, 15]. There exist two kinds of Clifford al-gebras, orthogonal and symplectic [16]. In orthogonal Clifford algebras, thesymmetric product of two basis vectors γa is the inner product and it gives theorthogonal metric, while the antisymmetric product gives a basis bivector:

γa · γb ≡ 12(γaγb + γbγa) = gab (orthogonal metric)

γa ∧ γb ≡ 12(γaγb − γbγa) (bivector) (1.1)

In symplectic Clifford algebras, the antisymmetric product of two basis vec-tors qa is the inner product and it gives the symplectic metric, whilst thesymmetric product gives a basis bivector:

qa ∧ qb ≡ 12(qaqb − qbqa) = Jab (symplectic metric)

qa · qb ≡ 12(qaqb + qbqa) (bivector) (1.2)

The generators of an orthogonal Clifford algebra can be transformedinto a basis (the Witt basis) in which they behave as fermionic creation andannihilation operators. The generators of a symplectic Clifford algebra be-have as bosonic creation and annihilation operators. We will show how bothkinds of operators can be united into a single structure so that they form abasis of a ‘superspace’. We will consider an action for a point particle in suchsuperspace. It contains the well known spinning particle action as a particu-lar case [17]. After quantization we obtain, in particular, the Dirac equation.Spinors can be constructed in terms of the orthogonal basis vectors rewrittenin the Witt basis. So a spinor of each minimal left ideal of an orthogonalClifford algebra is an element of the Fock space, whose basis elements areproducts of creation operators acting on a vacuum which in turn is the prod-uct of all annihilation operators [18]–[28]. The role of creation and annihila-tion operators together with the corresponding vacuum can be interchanged,and so we obtain [28] as many different vacua—and thus different kinds ofspinors—as there are different minimal left ideals of the Clifford algebra.This property has been exploited in order to explain the ‘internal’ degrees offreedom behind gauge theories of fundamental forces [29, 30, 31, 28]. Otherapproaches to the unification by Clifford algebras have been investigated inRefs. [32]–[46].

Instead of finite dimensional spaces, we can consider infinite dimen-sional spaces. Then we have description a field theory in terms of fermionicand bosonic creation and annihilation operators. The latter operators can beconsidered as being related to the basis vectors of the corresponding infinitedimensional space.

Our approach provides a fresh view on quantization. ‘Quantization’ isreplacement of the phase space coordinates with the corresponding basis vec-tors. The latter vectors are quantum mechanical operators. We point outthat, in the symplectic case, the Poisson brackets between phase space coor-dinates are equal to the wedge products (i.e., to 1

2 times the commutators)

Vol. 22 (2012) Quantized Fields Based on Cliff ord Algebras 451

of the corresponding basis vectors. We show that, for quadratic Hamiltoni-ans, the latter vectors satisfy the Heisenberg equations of motion, under theassumption that the classical trajectories in phase space are arbitrary, andnot necessarily solutions to the classical equations of motion. The analogousholds in the orthogonal case.

Such novel insight on the basis vectors, namely that they give the metricof spacetime (a subspace of a phase space), and at the same time they havethe role of quantum mechanical operators from which one can create, e.g.,spinors, could be exploited in the development of quantum gravity.

In Sec. 2 we review spaces with orthogonal and symplectic forms andtheir role in quantization. In Sec. 3 we present a relation between the clas-sical equations of motion and the Heisenberg equations for operators. Weconsider the latter relation for a generic ‘Hamiltonian’ that is quadratic incoordinates and momenta. In Sec. 4 we formulate a theory of a point particlein the space of commuting and anticommuting (Grassmann) coordinates, theaction being a generalization of the spinning particle action [17] and of the lo-cal Sp(2) symmetric action considered in Refs. [47]–[50]. In Sec. 5 we discussthe representation of spinors in terms of Grassmann coordinates. In Sec. 6we present a theory of quantized bosonic and fermionic fields in terms ofsymplectic and orthogonal Clifford algebras. We show that there exist manypossible definitions of vacua and fermionic creation and annihilation opera-tors, and how this fact could be exploited for resolution of the cosmologicalconstant problem. Finally, in Sec. 7 we discuss what are the prospects of suchtheory for grand unification and quantum gravity.

2. Spaces with Orthogonal an Symplectic Forms

2.1. The inner product and metric

I. Orthogonal caseThe inner product of vectors a and b is given by

(a, b)g = (aaγa, bbγb)g = aa(γa, γb)gbb = aagabbb. (2.1)

The metric is given by the inner product of two basis vectors:

(γa, γb)g = gab, (2.2)

where gab is a symmetric tensor. For basis vectors we can take the generatorsof the orthogonal Clifford algebra satisfying

(γa, γb)g = γa · γb ≡ 12(γaγb + γbγa) = gab. (2.3)

Here vectors are Clifford numbers. The inner product of the vectors a and bis given by the symmetric part of the Clifford product

(a, b)g =12(ab + ba) = a · b. (2.4)


II. Symplectic caseThe inner product of vectors z and z′ is given by

(z, z′)J = (zaqa, z′bqb)J = za(qa, qb)Jz′b = zaJabz′b, (2.5)

where(qa, qb)J = Jab (2.6)

is the symplectic metric.For a symplectic basis we can take generators of the symplectic Clifford

algebra:

(qa, qb)J = qa ∧ qb ≡ 12(qaqb − qbqa) = Jab (2.7)

Vectors are now symplectic Clifford numbers. The inner product of symplecticvectors z and z′ is given by the antisymmetric product

(z, z′)J =12(z z′ − z′z) = z ∧ z′. (2.8)

Dimension of the symplectic vector space is even. Physically, it is real-ized as the phase space. We can split a symplectic vector according to

z = zaqa = xμq(x)μ + pμq(p)μ , μ = 1, ..., n. (2.9)

Then we have

(z, z′)J = zaJabz′b = (xμp′ν − pνx′ν)gμν , (2.10)

with

Jab =(

0 gμν−gμν 0

), (2.11)

where, depending on signature, the n×n block gμν is the euclidean, gμν = δμν ,or the Minkowski metric, gμν = ημν .

Relations12[qa, qb] = Jab (2.12)

give12[q(x)μ , q(x)ν ] = 0 ,

12[q(p)μ , q(p)ν ] = 0, (2.13)

12[q(x)μ , q(p)ν ] = gμν , (2.14)

which are the Heisenberg commutation relations.

2.2. Poisson bracket

In symplectic case, the Poisson bracket between functions f(z) and g(z) ofphase space coordinates za is given by

{f, g}PB ≡ ∂f

∂zaJab ∂g

∂zb, (2.15)

where

Jab =(

0 − gμν

gμν 0

)(2.16)

is the inverse symplectic metric.


By introducing the symplectic basis vectors, we can rewrite the aboveexpression as the wedge product ∂f

∂za qa ∧ ∂g

∂zb qb. i.e.,

{f, g}PB =12

[∂f

∂zaqa,

∂g

∂zbqb] =

∂f

∂zaJab ∂g

∂zb(2.17)

If we take

f = zc , g = zd (2.18)

we obtain

{zc, zd}PB =12[qc, qd] = Jcd (2.19)

These are the Heisenberg commutation relations for ‘operators’ qc, qd.Usually by ‘quantization’ we mean the replacement of the classical phase

space coordinates za = (xμ, pμ) by operators za = (xμ, pμ) that satisfy theHeisenberg commutation relations. The above derivation reveals that thequantum mechanical operators za = (xμ, pμ) are in fact the symplectic basisvectors qa. This is true up to the factor i in front of the ‘momentum’ partqμ(p), a factor that is necessary in order to have Hermitian momentum oper-ator pμ. We also see that the Poison bracket of the phase space coordinatesis equal (apart from factor 1

2 ) to the commutator of the corresponding basisvectors.

In orthogonal case, the Poisson bracket between functions f(λ) and g(λ)of phase space coordinates λa is given by

{f, g}PB ≡ ∂f

∂λagab

∂g

∂λb, (2.20)

where

gab =(

gμν 00 gμν

)(2.21)

is the inverse orthogonal metric. In Sec. 4 we point out that in a phase spacewith orthogonal metric the coordinates λa must be Grassmann valued.

By introducing the basis vectors we can rewrite the above expression asthe dot product

(∂f∂λa γ

a)·(

∂g∂λb γ

b), i.e.,

{f, g}PB =12

{∂f

∂λaγa,

∂g

∂λbγb

}=

∂f

∂λagab

∂g

∂λb(2.22)

If we take

f = λc , g = λd, (2.23)

then we obtain

{λc, λd}PB =12{γc, γd} ≡ 1

2(γcγd + γdγc) = gcd (2.24)

These are the anticommutation relations for ‘operators’ γc, γd. The genera-tors γa of an orthogonal Clifford algebra are thus ‘quantized’ λa.


2.3. Representation of Clifford numbers

I. Orthogonal Clifford algebra Cl(2n)In even dimensions we can split the generators γa according to

γa = (γμ, γμ) , μ = 1, 2, ..., n (2.25)

We can introduce the Witt basis:

θμ = 1√2(γμ + i γμ)

θμ = 1√2(γμ − i γμ)

(2.26)

Then the anticommutation relations (2.3) become

θμ · θν ≡ 12 (θμθν + θνθμ) = ημν , θμ · θν = 0 , θμ · θν = 0. (2.27)

The generators γμ, γμ, θμ, θμ can be represented:1) as matrices,2) in terms of Grassmann coordinates:

θμ →√

2ξμ , θμ →√

2∂

∂ξμ(2.28)

ξμξν + ξνξμ = 0. (2.29)

II. Symplectic Clifford algebra ClS(2n)The generators are:

qa = (q(x)μ , q(p)μ ) , μ = 1, 2, ..., n (2.30)

Commutation relation (2.12) or (2.13),(2.14) can be represented:1) as 4 x 4 matrices, in which case the operators cannot be cast into the

Hermitian form,2) in terms of commuting coordinates:

qμ(x) →√

2xμ , q(p)μ →√

2∂

∂xμ(2.31)

xμxν − xνxμ = 0 (2.32)

3. Heisenberg Equations as Equations of Motion for BasisVectors

3.1. General considerations

Let us now consider a particle whose motion is described by the followingphase space action:

I =12

∫dτ (zaJabzb + zaKabz

b), (3.1)

where12zaKabz

b = H. (3.2)


Here Kab is a symmetric 2n × 2n matrix. By varying the latter action withrespect to za we obtain the equations of motion

za = Jab ∂H

∂zb. (3.3)

Let us consider trajectories za(τ) as components of an infinite dimen-sional vector

z = za(τ)qa(τ) ≡∫

dτ za(τ) qa(τ), (3.4)

where qa(τ) are basis vectors satisfying

qa(τ) ∧ qb(τ ′) = Ja(τ)b(τ ′) = Jab δ(τ − τ ′), (3.5)

and write the action (3.1) in the form

I =12za(τ)Ja(τ)b(τ ′)z

b(τ ′) +12za(τ)Ka(τ)b(τ ′)z

b(τ ′). (3.6)

We have introduced12

∫dτ za(τ)Kabz

b(τ) ≡ 12za(τ)Ka(τ)b(τ ′)z

b(τ ′) ≡ H, (3.7)

whereKa(τ)b(τ ′) = Kb(τ ′)a(τ) = Kabδ(τ − τ ′). (3.8)

The action (3.6) gives the following form of the equations of motion:

za(τ) = Ja(τ)b(τ ′) ∂H∂zb(τ ′) = Ja(τ)c(τ ′′) Kc(τ ′′)b(τ ′)z

b(τ ′), (3.9)

where ∂/∂za(τ) ≡ δ/δza(τ) denotes functional derivative. Multiplying bothsides of the latter equation by the basis vectors qa(τ) we obtain an equivalentequation

za(τ)qa(τ) = −qa(τ)Ka(τ)b(τ ′)zb(τ ′). (3.10)

Let us now use the following relation:

za(τ)qa(τ) ≡∫

dτ za(τ)qa(τ) = −∫

dτza(τ)qa(τ) ≡ −za(τ)qa(τ), (3.11)

that holds, if the boundary term vanishes, which we will assume is the case.Then Eq. (3.10) becomes

zb(τ′)qb(τ ′) = qa(τ)Ka(τ)b(τ ′)z

b(τ ′). (3.12)

Eq. (3.12) is equivalent to the classical equations of motion derived from theaction (3.1). It holds for a classical trajectory zb(τ

′) ≡ zb(τ ′) satisfying theminimal action principle.

Let us now explore what happens if we drop the requirement thatEq. (3.12) must hold for a trajectory that satisfies the minimal action princi-ple associated with (3.1), and make the assumption that Eq. (3.12) is satisfiedfor an arbitrary trajectory (path) zb(τ

′). Then we have

qb(τ ′) = qa(τ)Ka(τ)b(τ ′), (3.13)


which is an equation of motion for the basis vectors (operators) qa(τ) ≡ qa(τ).From Eqs. (3.8) and (3.13) we have

qb(τ) = qa(τ)Kab. (3.14)

The right hand side of the latter equation can be written as

Kab qb = [qa, H], H =

12qaKab q

b (3.15)

i.e., as the inner product (up to the factor 2) of a symplectic vector qa withthe symplectic bivector H. Remember that in the symplectic case the innerproduct is given by the commutator. So we obtain

qa = [qa, H], (3.16)

which are the Heisenberg equations of motion.We have found that the basis vectors of phase space satisfy the Heisen-

berg equations of motion. This is in agreement with the finding of Sec. 2 thatthe quantum mechanical operators za = (xμ, pμ) are in fact the symplecticbasis vectors qa. We have arrived at the Heisenberg equations (3.16) fromEq. (3.12) in which we assumed that zb(τ) ≡ zb(τ) was arbitrary. Arbitraryzb(τ) = (xμ(τ), pμ(τ)) means that coordinates and momenta are undeter-mined. In other words, if in the classical equations of motion (3.12) we as-sume that coordinates and momenta are undetermined, then we obtain theoperator equations (3.16). This sheds new light on ‘quantization’.

3.2. Particular physical cases

The general form of the action (3.1) contains particular cases that depend onchoice of Kab and the interpretation of coordinates xμ and parameter τ .

(i) One possibility is to interpret xμ, μ = 1, 2, ..., n, as coordinates of anon relativistic point particle, and τ as time t, the signature being (++++...)Then (3.1) describes a non relativistic harmonic oscillator in n-dimensions.

(ii) Another possibility is to interpret xμ, μ = 0, 1, 2, ..., n − 1, as co-ordinates in n-dimensional spacetime with signature (+ − − − ...), τ as anarbitrary monotonically increasing parameter along a particle’s worldline,take

Kab =(

0n×n 00 α(τ)ημν

), (3.17)

and assume that α is a Lagrange multiplier. Then we have a phase spaceaction for a massless relativistic particle in a higher dimensional spacetime:

I[xμ, pμ] = 12

∫dτ (xμημνp

ν−xμημν pν−αpμημνp

ν) =∫

dτ(pμx

μ − α

2pμp

μ)

(3.18)In a 4-dimensional subspace with signature (+−−−), such particle behavesas a massive relativistic particle.

More generally, we can take

Kab ≡ Kiμ

jν = Aijδ

μν =

(A1

1 A12

A21 A2

2

)δμν , (3.19)


and consider Aij(τ) as Lagrange multipliers that give three independent con-

straints. Then (3.1) becomes the Bars action [47]–[50]. Functions Aij(τ) have

the role of compensating gauge field that render the action (3.1) invariantunder local symplectic transformations of Sp(2). Here indices i, j = 1, 2, oc-curring in double indices a ≡ iμ, b ≡ jν, distinguish xμ from pμ. Some moreexplanation can be found in Sec. 4.3, where it is also shown that the Barsaction is a special case of a super phase space action.

4. Point Particle in ‘Superspace’

4.1. A generalized space spanned over an orthogonal and symplectic basis

Let us introduce the generalized vector space whose elements are:

Z = zAqA (4.1)where

zA = (za, λa) , za = (xμ, xμ) , λa = (λμ, λμ) (4.2)qA = (qa, γa) , qa = (qμ, qμ) , γa = (γμ, γμ) (4.3)

The scalar part of the product of two such basis elements gives themetric

〈qAqB〉S = GAB =(

Jab 00 gab

)(4.4)

where

qa ∧ qb = Jab =(

0 ημν−ημν 0

)(4.5)

γa · γb = gab =(

ημν 00 ημν

)(4.6)

Let us consider a particle moving in such space. Its worldline is givenby:

zA = ZA(τ) (4.7)where τ is a parameter on the worldline.

4.2. Examples of possible actions

A possible action is

I =12

∫dτ 〈ZAqAqBZ

B〉S =12

∫dτ ZAGABZ

B (4.8)

Inserting the metric (4.4) and the coordinates (4.2), we have

I =12

∫dτ (zaJabzb + λagabλ

b) (4.9)

Since Jab = −Jba and gab = gba, the first term differs from zero, if za areGrassmann (anticommuting) coordinates, whilst the second term differs fromzero if λa are commuting coordinates.


Another possible action, more precisely, a part of a phase space action,is

I =12

∫dτ 〈ZAqAqBZ

B〉S =12

∫dτ ZAGABZ

B (4.10)

or

I =12

∫dτ 〈zaqaqbzb + λaγaγbλ

b〉S =12

∫dτ (zaJabzb + λagabλ

b) (4.11)

Here, in order to have non vanishing terms, za must be commuting, and λa

Grassmann (anticommuting) coordinates.The corresponding canonical momenta are

p(z)a =∂L

∂za=

12Jabz

b, (4.12)

p(λ)a =∂L

∂λa=

12gabλ

a (4.13)

Inserting expressions (4.5) and (4.6) for Jab and gab, the action (4.11)becomes

I =12

∫dτ (xμημν x

ν − ˙xμημνx

ν + λμημνλν + ˙λ

μημν λ

ν) (4.14)

where[xμ, xν ] = 0, [xμ, xν ] = 0, (4.15){λμ, λν} = 0, {λμ, λν} = 0. (4.16)

The canonical momenta now read

p(x)μ =∂L

∂xμ=

12ημν x

ν , p(x)μ =∂L

∂xμ= −1

2ημνx

ν (4.17)

p(λ)μ =∂L

∂λμ=

12ημνλ

ν , p(λ)μ =∂L

∂λμ=

12ημν λ

ν (4.18)

Upon quantization, the coordinates and momenta become operators,

xμ, p(x)μ → xμ, p(x)μ (4.19)

λμ, p(λ)μ → λμ, p(λ)μ (4.20)satisfying

[xμ, p(x)ν ] = i δμν , [xμ, xν ] = 0 , [p(x)μ , p(x)ν ] = 0, (4.21)

{λμ, p(λ)ν } = i δμν , {λμ, λν} = 0 , {p(λ)μ , p(λ)ν } = 0 (4.22)Similar relations hold for barred quantities. Altogether, we have

za , p(z)a → za , p(z)a (4.23)

λa, p(λ)a → λa, p(λ)a (4.24)where the operators satisfy

[za, p(z)b ] = iδab (4.25)

{λa, p(λ)a } = iδab (4.26)If we insert

p(z)a =12Jabz

b, p(λ)a =12gabλ

a, (4.27)


we obtain12[za , zb] = i Jab,

12{λa, λb } = i gab. (4.28)

But we see that the above operator equations are just the relations for thebasis vectors of the orthogonal and symplectic Clifford algebras, providedthat we identify:

za = (qμ, iqμ), λa = (γμ, iγμ) (4.29)

We se that ‘quantization’ is in fact the replacements of the coordinatesza, λa with the corresponding basis vectors. The only difference is in thefactor i in front of qμ. Basis vectors, entering the action, are ‘quantum oper-ators’, apart from the i in the relations (4.29).

Instead of coordinates λa and basis vectors γa it is convenient to intro-duce new coordinates and new basis vectors

λ′ a ≡ ξa = (ξμ, ξμ) , ξμ = 1√2(λμ − i λμ)

ξμ = 1√2(λμ + i λμ)

(4.30)

γ′a ≡ θa = (θμ, θμ) , θμ = 1√

2(γμ + i γμ)

θμ = 1√2(γμ − i γμ)

(4.31)

In the new coordinates we have

g′ab = γ′a · γ′

b ≡ θa · θb =(

0 ημνημν 0

)(4.32)

and the action (4.11) becomes

I =12

∫dτ (zaJabzb + ξag′abξ

b) =12

∫dτ (zaJabzb + ξμημν ξ

ν + ˙ξμημνξ

ν)

(4.33)Then the canonical momenta are

p(x)μ =∂L

∂xμ=

12ημν x

ν , p(x)μ =∂L

∂ ˙xμ= −1

2ημνx

ν (4.34)

p(ξ)μ =∂L

∂ξμ=

12ημν ξ

ν , p(ξ)μ =∂L

∂ ˙ξμ=

12ημνξ

ν (4.35)

We see that now the canonically conjugate pairs are (ξμ, 12 ξμ) and (ξμ, 1

2ξμ).In old coordinates and basis vectors the situation was somewhat unfortunate,because the canonically conjugate variables were (λμ, 1

2λμ) and (λμ, 12 λμ),

i.e., the canonical momenta were essentially the same as the variables whichthey were conjugated to.

The commuting coordinates za = (xμ, xμ) ≡ (xμ, pμ) and the symplecticbasis vectors qa = (qμ, qμ) span a subspace that we will call the bosonicsubspace. The Grassmann (anticommuting) coordinates λa = (λμ, λμ) andthe orthogonal basis vectors γa = (γμ, γμ), or equivalently, the Grassmancoordinates ξa = (ξμ, ξμ) and the Witt basis vectors θa = (θμ, θμ), spana subspace that we will call the fermionic subspace. In Sec. 2.3 we pointed


out that (qμ, qμ) can be represented by (xμ, ∂/∂xμ), whilst (θμ, θμ) can berepresented by (ξμ, ∂/ξμ).

4.3. Completing the super phase space action

The action considered in the previous subsection is not complete. An addi-tional term is needed. Let the τ -derivative in the action

I =12

∫dτ ZAGABZ

B (4.36)

be replaced with the covariant derivative

ZA → ZA + AACZ

C , (4.37)

where ABC depend on τ . So we obtain

I =12

∫dτ (ZA + AA

CZC)GABZ

B (4.38)

which is invariant under τ -dependet, i.e., local, ‘rotations’ of ZA, and whereAA

C(τ) are the corresponding compensating gauge fields. We will take intoaccount only particular local ‘rotations’ of ZA, namely those that rotate be-tween the canonically conjugate pairs of the (super) phase space variablesZA. In the bosonic subspace the latter local ‘rotations’ manifest themselvesas local symplectic transformations of Sp(2). Then the corresponding com-pensating gauge fields are Aiμ

jν = Aijδ

μν , where the i, j = 1, 2 distinguishes

xμ from xμ, since za can be written in the form za = ziμ ≡ (xμ, xμ). Thereare only three independent gauge fields Ai

j(τ); they represent a choice ofgauge, as discussed in refs [47]–[50], and have thus the role of Lagrange mul-tipliers. Analogous considerations hold in the fermionic subspace. No kineticterm for the gauge fields AA

C so designed is necessary.The action (4.38) is the Bars action [47]–[50], generalized to the super-

space (see also Ref. [51]). In particular, the extra term gives

AACZ

CGABZB = αpμp

μ + β λμpμ + γλμpμ, (4.39)

where α, β, γ are Lagrange multipliers contained in AAC . Other choices

of Lagrange multipliers AAC are possible, and they give expressions that

are different from Eq. (4.39). For the bosonic subspace, this was discussed inRefs. [47]–[50].

For the case (4.39), the action (4.38) gives the following constraints:

pμpμ = 0 , λμpμ = 0 , λμpμ = 0. (4.40)

Mass comes from extra dimensions. It was shown by Bars [47]–[50] that theconsistency of the constraints resulting from the action (4.38) requires oneextra time like and one extra space like dimension of the bosonic subspace.Consequently, also the dimensionality of the fermionic subspace is adequatelyenlarged. So in this theory there are in fact extra dimensions that allow forthe occurrence of mass in four dimensions.


Upon quantization, the classical constraints λμpμ = 0 and λμpμ = 0become two copies of the Dirac equation

ˆλμpμΨ = 0, and λμpμΨ = 0 (4.41)

whereλμ = γμ , and ˆλμ = iγμ. (4.42)

The state Ψ can be represented:1) as a column ψα(x),2) as a function ψ(xμ, ξμ).

The λμ = γμ can be represented1) as matrices (γμ)αβ ,2) as ξμ + ∂

∂ξμ .

We also have ˆλμ = i γμ, which can be represented1) as matrices i(ˆγμ)αβ ,2) as i (ξμ − ∂

∂ξμ ).where ξμ are Grassmann coordinates (see Eq.(4.30)).

4.4. Building up spinors from basis vectors

We have seen that upon quantization the classical Grassmann coordinatesλa = (λμ, λμ) become operators λa = (λμ, ˆλμ) = (γμ, iγμ), where γμ, γμ aregenerators (basis vectors) of Cl(2n), or in general of Cl(p, q), p+ q = 2n. Inthe Witt basis (2.26) the basis vectors satisfy the fermionic anticommutationrelations (2.27). Using the basis vectors θμ, θμ we can build up spinors bytaking a ‘vacuum’

Ω =∏μ

θμ that satisfies θμ Ω = 0 (4.43)

and acting on it by the ‘creation’ operators θμ. So we obtain a ‘Fock space’basis for spinors, that contains 2n independent elements:

sα = (1Ω, θμΩ , θμθνΩ , θμθνθρΩ , θμθνθρθσΩ, ...) (4.44)

in terms of which any state can be expanded:

ΨΩ =∑

ψαsα (4.45)

With operators θμ, θμ, defined above, we can construct spinors as the el-ements of a minimal left ideal of Cl(p, q), p + q = 2n, where n is dimen-sion of spacetime. Notice that in the case of 4-dimensional spacetime, i.e.,when n = 4, the symplectic or bosonic phase space is 8-dimensional. There-fore, if with every commuting phase space coordinate za = (xμ, xμ) oneassociates a corresponding Grassmann phase space coordinate λa = (λμ, λμ)(i.e., ξa = (ξμ, ξμ), in the Witt basis), then also the ‘orthogonal’ or fermionicphase space is 8-dimensional. In the usual theory of the spinning particle,only four Grassmann coordinates λμ (or ξμ) are considered, which then leads


to the spinors of Cl(1, 3). The usual theory thus imposes an asymmetry be-tween the bosonic phase space, which has eight dimensions, and the fermionicphase space, which has four dimensions.

Definition (4.43) is just one of the possible definitions of vacuum. Ingeneral, vacuum can be defined as the product of n factors, some of whichare θμ and some are θμ:

Ω = θμ1θμ2 ...θμr θμr+1 θμr+2 ...θμn , r = 0, 1, 2, ..., n (4.46)

The number of different vacua, 2n, is equal to the number of left minimalideals of Cl(2n) (more precisely, Cl(p, q), p+ q = 2n). With each vacuum soconstructed, we can associate a different Fock space basis for spinors of thecorresponding left ideal. The direct sum of all those different spinor spaces isthe Clifford algebra Cl(p, q), p + q = 2n, whose generic element, denoted Ψ,can be expanded according to

Ψ = ψαβsαβ ≡ ψAsA. (4.47)

Here the first index, α, denotes the spinor components of a given ideal, whichis denoted by the second index, β. It is convenient to denote the double indexαβ by a single index A. In the case, when we start from the 4-dimensionalMinkowski spacetime with signature (1, 3), the signature of phase space is(2, 6), and we obtain Cl(2, 6).

With the Witt basis (4.31), we obtain the spinor spaces as subspaces ofCl(2, 6), but those spinor spaces do not contain the ordinary spinors of M1,3.The latter spinors are constructed [25, 28] in terms of a different Witt basis,which contains the elements

θ1 = 1√2(γ0 + γ3), θ2 = 1√

2(γ1 + i γ2)

θ1 = 1√2(γ0 − γ3), θ2 = 1√

2(γ1 − iγ2).

(4.48)

Then the full basis of the 8-dimensional phase space contains, in addition tothe above elements, also the following elements:

θ3 = 1√2(γ0 + γ3), θ4 = 1√

2(γ1 + i γ2)

θ3 = 1√2(γ0 − γ3), θ4 = 1√

2(γ1 − i γ2).

(4.49)

Taking the scalar product of Ψ with the basis elements, we obtain thecomponents:

〈sA‡Ψ〉S = ψA, (4.50)

where 〈〉S denotes the scalar part of a Clifford algebra valued object. Op-eration ‡ reverses the order of vectors in a Clifford product and performscomplex conjugation. More details can be found in Ref. [31].

Components ψA represent a generic element Ψ of the Clifford algebra.If we take the scalar product of Ψ with basis elements of one left ideal, say,α1 ≡ α,

〈sα‡Ψ〉S = ψα, (4.51)


then the components ψα represent a spinor state of this chosen left ideal.A state Ψ can be represented by its components ψA, which are projec-

tions of Ψ onto the basis elements. Alternatively, since it is a Clifford algebravalued object, it can be represented by matrices

〈sA‡Ψ sB〉S = (Ψ)AB . (4.52)

These are 22n × 22n matrices, and they belong to a reducible representationof the Clifford algebra1 Cl(2n). If instead of the full basis sA ≡ sαβ , we takethe basis elements of a chosen left ideal, e.g., sα1 ≡ sα, then we have

〈sα‡Ψ sβ〉S = (Ψ)αβ = ψA〈sαsA sβ〉S = ψA(sA)αβ. (4.53)

This shows how every element of a Clifford algebra can be represented as anirreducible 2n × 2n matrix. For instance, γμ can be represented by a matrix(γμ)αβ , and θμ by (θμ)αβ .

5. On the Representation of Spinors in Terms of theGrassmann Coordinates

Operators θμ, θμ, which are in fact nothing but the generators of the Cliffordalgebra, expressed in the Witt basis (2.26) or (4.31), can be represented eitheras matrices or in terms of the Grassmann coordinates and their derivatives:

θμ →√

2ξμ , θμ →√

2∂

∂ξμ(5.1)

A spinor state is then represented by a wave function ψ(ξμ). Since ξμ areGrassmann, anticummuting, coordinates, the Taylor expansion of ψ(ξμ) hasa finite number of terms, namely 2n, which is the same as the number ofcomponents of a spinor in an 2n-dimensional space.

The definition of vacuum θμΩ = 0, μ = 1, 2, ..., n, is now representedby the equation

∂

∂ξμΩ(ξμ) = 0 (5.2)

whose solution is a constant, e.g., Ω(ξμ) = 1. A state (4.45) can then berepresented as

ΨΩ → ψ(ξμ) =r=n∑r=0

ψμ1μ2...μrξμ1ξμ2 ...ξμr . (5.3)

Some other definition of vacuum, e.g., θ1θ2θ3θ4....θnΩ = 0, is represented bythe equation

ξ1ξ2∂

∂ξ3∂

∂ξ4....

∂

∂ξnΩ(ξμ) = 0, (5.4)

1For the sake of simplicity, we will use the simplified notation Cl(2n) for Cl(p, q), p+ q =2n.


which has for a solution Ω(ξμ) = ξ1ξ2. The corresponding state is then rep-resented as

ΨΩ → ψ(ξμ) =

(r=n∑r=0

ψμ1μ2...μrξμ1ξμ2 ...ξμr

)ξ1ξ2 . (5.5)

In general, a vacuum state is given by (4.46), which can be represented byfunction

Ω(ξμ) = ξμ1ξμ2 ...ξμs , (5.6)

and the corresponding spinor state is represented as

ΨΩ → ψ(ξμ) =

(r=n∑r=0

ψμ1μ2...μrξμ1ξμ2 ...ξμr

)ξν1ξν2 ...ξνs . (5.7)

The spinor states of every minimal left ideal of Cl(p, q), p + q = n, canbe thus represented in terms of a function of n Grassmann coordinates ξμ,μ = 1, 2, ..., n.

So far we have considered functions of Grassmann coordinates ξμ. Thosefunctions can be considered as components of vectors (states) ΨΩ with respectto a basis, say h(ξ):

ΨΩ =∫

dξ1dξ2...dξnψ(ξ)h(ξ). (5.8)

This can be written in a more compact notation as

ΨΩ = ψ(ξ)h(ξ), (5.9)

where ψ(ξ) ≡ ψ(ξ) ≡ ψ(ξμ), and h(ξ) ≡ h(ξ) ≡ h(ξμ). Here (ξ) is written asan index denoting a component of the vector ΨΩ, and if the index is repeated,then the integration over ξ in the sense of Eq. (5.8) is implied. Componentsψ(ξ) can be complex valued; then also the basis vectors are complex (i.e.,consisting of two real components, as described in more detail later).

Basis vectors are assumed to satisfy

h(ξ).h(ξ′) = 12 (h(ξ)h(ξ′) + h(ξ′)h

(ξ)) = δ(ξ)(ξ′) (5.10)

where δ(ξ)(ξ′) ≡ δ(ξ − ξ′) is the delta ‘function’ in the Grassmann space,satisfying

δ(ξ)(ξ′)f(ξ′) = f (ξ) (5.11)

Indices are lowered and raised, respectively, by the metric ρ(ξ)(ξ′) = h(ξ).h(ξ′)

and its inverse ρ(ξ)(ξ′). If h(ξ) are complex, then h(ξ) actually means hi(ξ) =

(h1(ξ), h2(ξ)) or hi(ξ) = (h (ξ), h∗(ξ)). Analogously, h(ξ) means hi(ξ) =(h1(ξ), h2(ξ)) or hi(ξ) = (h (ξ), h∗(ξ)). Metric ρ(ξ)(ξ′) thus has an implicit indexi = 1, 2.


In Eq. (5.9) we can perform the expansion in terms of ξμ, and use theproperties

∫dξ = 0,

∫dξ ξ = 1, which leads to

ψ(ξ)h(ξ) =∫

dnξ

(ψ

∣∣∣∣0

+∂ψ

∂ξμ

∣∣∣∣0

ξμ + ... +∂nψ

∂ξμ1 ...∂ξμn

∣∣∣∣0

ξμ1 ...ξμn

)

×(h

∣∣∣∣0

+∂h

∂ξμ

∣∣∣∣0

ξμ + ... +∂nh

∂ξμ1 ...∂ξμn

∣∣∣∣0

ξμ1 ...ξμn

)

= ψ

∣∣∣∣0

∂nh

∂ξ1...ξn

∣∣∣∣0

+∂ψ

∂ξμ

∣∣∣∣0

∂n−1h

∂ξ1... ˆ∂ξμ...∂ξn

∣∣∣∣0

+... +∂nψ

∂ξ1...ξn

∣∣∣∣0

h

∣∣∣∣0

= ψαhα, α = 1, 2, ..., 2n. (5.12)

Here hα are discrete basis vectors spanning the 2n-dimensional space C2n in

which spinors of one minimal left ideal live. The basis vectors hα ∈ C2n are

used here instead of the basis spinors sα ∈ Cl(2n) defined in Eq. (4.44).One can get the contact with the usual language of quantum theory by

the correspondence h(ξ) = |ξ〉, h(ξ) = 〈ξ|, and hα = |α〉, hα = 〈α|. Then

ΨΩ = ψ(ξ)h(ξ) =∫

|ξ〉dnξ 〈ξ|ΨΩ〉 =∑α

|α〉〈α|ξ〉dnξ 〈ξ|ΨΩ〉 = hαcα(ξ)ψ

(ξ)

(5.13)The transformation coefficients cα(ξ) ≡ 〈α|ξ〉 can be read from Eq. (5.12).

Let us now consider a generic element Ψ ∈ Cl(2n). If we project Ψ ontothe basis h(ξ), then we obtain the wave function of Grassmann coordinates,

h(ξ).Ψ = h(ξ).(ψ(ξ′)h(ξ′)) = ψ(ξ). (5.14)

This is one possible representation of a state Ψ that is analogous to (4.51).In fact, it is a projection of Ψ onto one ideal. In analogy with Eq. (4.53), wecan put Ψ into a sandwich between h(ξ) and h(ξ), and consider matrices

〈h(ξ) Ψh(ξ′)〉S = (Ψ)(ξ)(ξ′) (5.15)

According to Eqs. (4.44),(4.45), a state Ψ is expressed in terms of basisvectors θμ, θμ, which, in turn can be expressed as linear combinations ofγμ, γμ (generators of Cl(2n)). Therefore, the following matrix elements areof particular interest:

θμ → 〈h(ξ) θμ h(ξ′)〉S = ξμ δ(ξ − ξ′)

θμ → 〈h(ξ) θμ h(ξ′)〉S =∂

∂ξμδ(ξ − ξ). (5.16)

They can be used as basic blocks for building up a matrix (5.15) that rep-resents a generic state Ψ, spanned over a spinor basis of all 2n minimal leftideals of Cl(2n).

Whereas a matrix (Ψ)(ξ)(ξ′) can represent any element Ψ of Cl(2n),components ψ(ξ) represent a spinor ΨΩ of one ideal only. From a 1st rankspinor, i.e., a spinor of one left minimal ideal,

ΨΩ = ψαhα = ψ(ξ)h(ξ), (5.17)


and its reverse2, interpreted as a spinor of a right minimal ideal

Ψ‡Ω = ψ∗αh‡

α = ψ∗(ξ)h‡(ξ), (5.18)

we can pass to a 2nd rank spinor by taking the tensor product

ΨΩ ⊗ Ψ′‡Ω = ψαψ

′∗βhα ⊗ h‡β = ψ(ξ)ψ

′∗(ξ′)h(ξ) ⊗ h‡(ξ′). (5.19)

Once we have the bases hα⊗h‡β and h(ξ)⊗h‡

(ξ′), we can span over them

the space of objects whose components are ψαβ and ψ(ξ)(ξ′), respectively.Components ψαβ represent a generic element of Cl(2n). Similarly, also com-ponents ψ(ξ)(ξ′) represent a generic element of Cl(2n). Instead of the doubleindices we can use the single indices, and write ψαβ ≡ ψA (as we did inEq. (4.47)), or ψ(ξ)(ξ′) ≡ ψ(ξ,ξ′). Instead of the tensor product basis hα ⊗ h‡

β

we can take the basis sαβ ≡ sA, used in Eq. (4.47)), and instead of h(ξ)⊗h‡(ξ′)

we can take another basis, denoted h(ξ)(ξ′) ≡ h(ξ,ξ′), which spans the spaceof Grassmann functions. An element of the latter space is

Ψ = ψ(ξ,ξ′)h(ξ,ξ′) (5.20)

Whereas a spinor of one left ideal is described by a function ψ(ξ) ≡ ψ(ξ) ofn Grassmann coordinates ξ ≡ ξμ, μ = 1, 2, ..., n, a generic element of Cl(2n)is described by a function ψ(ξ,ξ′) ≡ ψ(ξ, ξ′) of 2n Grassmann coordinates(ξ, ξ′) ≡ (ξμ, ξ′μ), μ, ν = 1, 2, ..., n. If we perform an expansion, analogous tothat of Eq. (5.12), we obtain 22n independent components.

In general, a wave function depends on commuting coordinates xμ aswell, because it has to form a representation, not only of the orthogonalbasis vectors (γμ, γμ) or (θμ, θμ), but also of the symplectic basis vectors(qμ, qμ) ≡ (q(x)μ , q

(p)μ ) according to Eq. (2.31). Therefore, instead of Eq. (5.9),

we haveΨΩ = ψ(x,ξ)h(x,ξ), (5.21)

whereh(x,ξ) · h(x′,ξ′) = δ(x,ξ)(x′,ξ′). (5.22)

A state ΨΩ is now an element of an infinite dimensional space spanned overa basis h(x,ξ), the components being a wave function ψ(x,ξ) ≡ ψ(x, ξ). In thenext section we will formulate foundations of the field theory based on theorthogonal and symplectic Clifford algebras.

6. Description of Fields

6.1. Bosonic fields

In Sec. 3, Eq. (3.1), we considered the phase space action, I[za], za = (xμ, pμ),of a point particle. The latter action is a functional of coordinates and mo-menta. If, by the equations of motion, the momenta can be expressed interms of coordinates, then we can obtain an equivalent action, I[xμ], which

2We define reversion so to include complex conjugation ‘*’.


is a functional of coordinates only. For instance, instead of the phase spaceaction (3.18), we obtain

I[xμ] = 12

∫dτ

xμxμ

λ. (6.1)

We can interpret the coordinates xμ, μ = 1, 2, ..., n, in several differentways, as we did in Sec. 3.2. For instance, the coordinates xμ can be interpretedto denote position (of a non relativistic particle) in n-dimensional space, Rn.The latter space is a vector space, spanned over the set of n-basis vectors,γμ, that can be generators of the Clifford algebra Cl(n), satisfying3

γμ · γν ≡ 12 (γμγν + γνγμ) = δμν , (6.2)

so that the action (6.1) reads

I[xμ] = 12

∫dτ

xμγμγν xν

λ(6.3)

Position in Rn is thus described by a vector x = xμγμ of the orthogonal

Clifford algebra Cl(n). Instead of the configuration space, and an actionI[xμ], such as (6.1), we can consider the corresponding phase space, whosepoints are described by symplectic vectors z = zaqa, considered in Sec. 2, theaction I[xμ, pμ] being, e.g., Eq. (3.18) or, in general, Eq. (3.1).

Analogously, we can consider a state vector Φ = φ(x)h(x), spanned overan infinite set of basis vectors h(x) that satisfy the relations of a generalizedorthogonal infinite dimensional Clifford algebra

h(x) · h(x′) = δ(x)(x′) . (6.4)

The latter vector is an infinite dimensional analog of a vector x = xμγμconsidered in previous paragraph. Equations of “motion” for φ(τ, x) ≡ φ(x)(τ)can be derived from an action functional

I[φ(τ, x)] =∫

dτ L(φ, ∂τφ, ∂μφ), (6.5)

which can be any one known in the field theory, e.g., the scalar field action

I[φ] = 12

∫dτ dnx

((∂τφ)2 + ∂μφ∂μφ−m2φ2

). (6.6)

where xμ are coordinates of space Rn, ∂τ ≡ ∂/∂τ , and ∂μ ≡ ∂/∂xμ. The

latter action can be written in the form

I[φ] = 12

∫dτ

(∂τφ

(x)h(x)h(x′)∂τφ(x′) + ∂μφ

(x)h(x)h(x′)∂μφ(x′)

−m2φ(x)h(x)h(x′)φ(x′)

), (6.7)

which is an infinite dimensional analog of the action (6.3).

3Alternatively, xμ can be interpreted as denoting position (of a relativistic particle) in

n-dimensional spacetime, Mn ≡ R(1,n−1), with signature (1, n− 1). Then, instead of δμν ,

we would have the Minkowski metric, ημν .


Introducing the momentum Π(x) = δL/δφ = φ and the HamiltonianH =

∫dnx (Π(τ, x)φ(τ, x) − L) we obtain the phase space action

I[φ,Π] =∫

dτ[∫

dnxΠφ−H

]

= 12

∫dτ dnx

[Πφ− Πφ− (Π2 − ∂μφ∂μφ + m2φ2)

]. (6.8)

The above equations also hold for complex fields, if an implicit index, c = 1, 2,denoting, e.g., the real and imaginary component, is assumed, with under-standing that φ2 ≡ φcφc = φ1φ1 + φ2φ2, or φ2 ≡ φcφc = 1

2 (φ∗φ + φφ∗).The action (6.8) is a particular case of a generic phase space action for

bosonic fields that we will consider in the following.

6.1.1. A generic phase space action for bosonic fields and its quantization.Let us consider a vector Φ = φi(x)ki(x), i, j = 1, 2, in an infinite dimen-sional space, and assume that the basis vectors ki(x) satisfy the relations ofa symplectic Clifford algebra:

ki(x) ∧ kj(x′) ≡ 12 (ki(x)kj(x′) − kj(x′)ki(x)) = Ji(x)j(x′). (6.9)

The symplectic metric has the following form:

Ji(x)j(x′) =(

0 δ(x)(x′)−δ(x)(x′) 0

), (6.10)

where δ(x)(x′) ≡ δ(x− x′). Shortly, Ji(x)j(x′) = εijδ(x− x′), with εji = −εij .Components φi(x) = (φ1(x), φ2(x)) ≡ (φ(x),Π(x)) are analogous to za =

(xμ, pμ) of Secs. 2 and 3, i.e., to coordinates and momenta.The symplectic vector reads explicitly

Φ = φi(x)ki(x) = φ1(x)k1(x) + φ2(x)k2(x) = φ(x)kφ(x) + Π(x)kΠ(x) (6.11)

The action is now

I = 12

∫dτ

(φi(x)Ji(x)j(x′)φ

j(x′) + φi(x)Ki(x)j(x′)φj(x′)

), (6.12)

where the fields φi(x) are assumed to be functions of τ , so that φi(x) meansφi(x)(τ) ≡ φi(τ, x). It gives the following equations of motion:

φi(x) = J i(x)j(x′) ∂H

∂φj(x′) , (6.13)

where

H = φi(x)Ki(x)j(x′)φj(x′) ≡

∫dnx dn x′φi(x)Kij(x, x′)φj(x′), (6.14)

and ∂/∂φj(x′) ≡ δ/δφj(x′). If we now follow the analogous procedure as inEqs. (3.6)–(3.16), we arrive at the equations of motion for the operators:

kj(x′) = ki(x)Ki(x)j(x′) = [kj(x′), H], (6.15)

whereH = 1

2ki(x)Ki(x)j(x′)k

j(x′). (6.16)


The latter operator equation of motion can be derived from the action

I = 12

∫dτ

(ki(x)Ji(x)j(x′)k

j(x′) + ki(x)Ki(x)j(x′)kj(x′)

). (6.17)

We obtain

ki(x) = {ki(x), H}P.B. =∂ki(x)

∂km(x′) Jm(x′)n(x′′) ∂H

∂kn(x′′)

= J i(x)n(x′′) ∂H

∂kn(x′′) = [ki(x), H]. (6.18)

In the above calculation we have assumed that ∂ki(x)/∂km(x′) = δi(x)m(x′).Indices are lowered and raised by the symplectic metric Ji(x)j(x′) and itsinverse J i(x)j(x′).

6.1.2. Some particular cases. The general form of the action (6.12) or itsquantum version (6.17) contains particular cases that depend on choice ofKi(x)j(x′), and on the space the coordinates x are associated with. Let usconsider some of the cases:

(i) For instance, let x ≡ xμ, μ = 1, 2, 3, be three spatial coordinates,and τ = t the non relativistic time. Instead of xμ, we will now write xr, r =1, 2, 3, so to have a closer contact with the conventional notation.

a) If we take

Ki(x)j(x′) =(

(m2 + ∂r∂r)δ(x− x′) 00 δ(x− x′)

), (6.19)

then the action (6.12) becomes the scalar field phase space action (6.8).b) If we take

Ki(x)j(x′) =(− 1

2m∂r∂r + V (x)

)δ(x− x′) gij , gij =

(0 11 0

)(6.20)

then the action (6.12) becomes

I=∫

dt d3x

[12 φ

i(t, x)εijφj(t, x) + iφi(t, x)(− 1

2m∂r∂r + V (x)

)gijφ

j(t, x)],

(6.21)where φi(t, x) = (φ(t, x),Π(t, x)) = (φ(t, x), iφ∗(t, x)). In Eq. (6.21) we havethe action for the classical Schrodinger field.

Similarly, using (6.17), the action for operators becomes

I =∫

dt d3x

[12 k

i(x)εijkj(x) + iki(x)(− 1

2m∂r∂r + V (x)

)gijk

j(x)],

(6.22)which is the action for the quantized Schrodinger field.

(ii) Alternatively, let x ≡ xμ, μ = 0, 1, 2, 3, be four coordinates ofspacetime, and τ a Lorentz invariant evolution parameter. Then the choice

Ki(x)j(x′) =(− 1

2Λ∂μ∂μ

)gijδ(x− x′) , gij =

(0 11 0

)(6.23)


for a constant Λ, gives the Stueckelberg action [52]–[61]

I =∫

dτ d4x

[12 φ

i(x)εijφj(x) + iφi(x)(− 1

2Λ∂μ∂μ

)gijφ

j(x)], (6.24)

and the analogous action for the quantized field ki(x).

6.2. Fermionic fields

Let us now consider the vector Ψ = ψi(x)hi(x), i = 1, 2, and assume that thebasis vectors hi(x) satisfy the relations of an orthogonal Clifford algebra

hi(x) · hj(x′) = 12 (hi(x)hj(x′) + hj(x′)hi(x)) = ρi(x)j(x′), (6.25)

where ρi(x)j(x′) is an orthogonal metric. Its explicit form depends on a chosenbasis. In a particular basis the metric can be

ρ′i(x)j(x′) = δijδ(x)(x′) =(δ(x)(x′) 0

0 δ(x)(x′)

). (6.26)

This can be transformed into another basis, namely the Witt basis, in which

ρi(x)j(x′) =(

0 δ(x)(x′)δ(x)(x′) 0

). (6.27)

We assume that an implicit spinor index, α = 1, 2, ..., 2n, occurs inthe expressions. Thus, Ψ = ψi(x)hi(x) ≡ ψiα(x)hiα(x). Components ψi(x) =(ψ1(x), ψ2(x)) ≡ (ψ(x), π(x)) are Grasmann valued phase space variables, andare analogous to ξa ≡ ξiμ = (ξμ, ξμ) considered in Sec. 4.

The fermionic field action that corresponds to the bosonic field action(6.12) is

I = 12

∫dτ

(ψi(x)ρi(x)j(x′)ψ

j(x′) + ψi(x)Hi(x)j(x′)ψj(x′)

), (6.28)

If we now repeat a procedure, analogous to that in Eqs. (6.12)–(6.18), thenwe obtain the following equations of motion for the basis vectors

hi(x) = {hi(x), H}, (6.29)

where H = hi(x)Hi(x)j(x′)hj(x′), and brace means the anticommutator.

Let, in particular, be x ≡ xr, r = 1, 2, 3, and τ = t. If ρi(x)j(x′) is givenby Eq. (6.27), and

Hi(x)j(x′) =(

0 −(α∗r∂r + im)δ(x− x′)(αr∂r + im)δ(x− x′) 0

), (6.30)

where αr = γ0γr are hermitian matrices in the spinorial indices, i.e., (αr)∗βα =(αr)αβ , then the action (6.28) becomes

I =∫

dt d3x[12 (πψ − πψ) + πγ0γr∂rψ −mπψ

], (6.31)

where we have taken into account the anticommutativity of ψ and π. If wewrite π = iψ† and omit the total derivative (d/dt)(πψ), we obtain the usual


phase space action for the Dirac field. The latter field can be the usual 2n/2-component Dirac field in n-dimensional spacetime, or it can be a field with2n components, considered in Secs. 4.4 and 5.

Instead of a discrete spinor index α we can take the Grassmann coor-diantes ξ and write the vector Ψ in the form Ψ = ψi(x,ξ)hi(x,ξ), where thebasis vectors satisfy

hi(x,ξ) · hj(x′,ξ′) = ρi(x,ξ)j(x′,ξ′) = δijδ(x)(x′)δ(ξ)(ξ′). (6.32)

If, as a model, we take two Grassmann coordinates only, ξ ≡ (ξ1, ξ2), thenΨ is a usual 4-component spinor (see Eq. (4.48)). A generic fermionic fieldaction can be written as

I = 12

∫dτ

(ψi(x,ξ)ρi(x,ξ)j(x′,ξ′)ψ

j(x′,ξ′) + ψi(x,ξ)Hi(x,ξ)j(x′,ξ′)ψj(x′,ξ′)

).

(6.33)In particular, for Hi(x,ξ)j(x′,ξ′) we can take a matrix, analogous to (6.30),

in which objects γμ are represented in terms of (ξ ± ∂/∂ξ)δ(ξ − ξ′).We can consider, for instance, the case (4.48), and represent

θ1 → ξ1δ(ξ − ξ′), θ2 → ξ2δ(ξ − ξ′), θ1 → ∂

∂ξ1δ(ξ − ξ′), θ2 → ∂

∂ξ2δ(ξ − ξ′),

(6.34)and invert the relations (4.48) so to express γμ in terms of θ1, θ2, θ1, θ2.Then the action (6.33) is equivalent to the usual action for the 4-componentspinor field.

Alternatively, we can consider the case (4.31), in which the numberof vectors, spanning the fermionic phase space is the same as the num-ber of vectors spanning the bosonic phase space, namely eight. Then ξ ≡ξμ, μ = 0, 1, 2, 3, and a vector Ψ = ψi(x,ξ)hi(x,ξ), for a fixed i, represents a16-component spinor field.

6.3. Poisson brackets

In Sec. 2.2 we have observed that in the symplectic case the Poisson bracketof the phase space coordinates is equal to the wedge product (i.e., to onehalf times the commutator) of the corresponding symplectic basis vectors.Similarly, in the orthogonal case, the Poisson bracket of the phase spacevariables is equal to one half of the anticommutator of the correspondingorthogonal basis vectors. Analogous holds for fields.

In symplectic case we have{f(φi(x)), g(φj(x))

}PB

=∂f

∂φi(x)J i(x)j(x′) ∂g

∂φj(x′)

=∂f

∂φi(x)12 [ki(x), kj(x

′)]∂g

∂φj(x′) (6.35)

In particular, if f(φi(x)) = φk(x′′) and g(φj(x)) = φ�(x′′′), then

{φk(x′′), φ�(x′′′)}PB = Jk(x′′)�(x′′′) = 12 [kk(x

′′), k�(x′′′)] (6.36)


The Poisson bracket of two classical fields is equal to the symplectic metricwhich, in turn, is equal to the wedge product (i.e., 1/2 times the commutator)of two symplectic basis vectors. The latter vectors are bosonic field operators,and from Eq. (6.36) we see that the canonical commutation relations are infact the relations (6.9) of a symplectic Clifford algebra.

In orthogonal case we have{f(ψi(x)), g(ψi(x))

}PB

=∂f

∂ψi(x)ρi(x)j(x

′) ∂g

∂ψj(x′)

=∂f

∂ψi(x)12{hi(x), hj(x′)} ∂g

∂ψj(x′) (6.37)

In particular, if f(ψi(x)) = ψk(x′′) and g(ψi(x)) = ψ�(x′′′), then

{φk(x′′), φ�(x′′′)}PB = ρk(x′′)�(x′′′) = 1

2{kk(x′′), k�(x

′′′)} (6.38)

The Poisson bracket of two classical fields is equal to the orthogonal metric.On the other hand, the orthogonal metric is equal to the symetrized product(given by the anticommutator) of two orthogonal basis vectors. The lattervectors are fermionic field operators, and Eq. (6.38) shows that the canonicalanticommutation relations for fermionic fields are in fact the relations (6.25)of an orthogonal Clifford algebra.

Spinor indices α, β are not explicitly displayed in Eqs. (6.37),(6.38).Another possibility is to rewrite the latter equations by replacing ψi(x) ≡ψiα(x) with ψi(x,ξ).

6.4. Generalization to ‘superfields’

Both actions (6.12),(6.28) can be unified into a single action by introducinga ‘superfield’

Ψ = ψAhA (6.39)

where ψA = (φi(x), ψi(x)) and hA = (ki(x), hi(x)). So we have

I[ψA] = 12

∫dτ(ψAGABψ

B + ψAHABψB) (6.40)

where

〈hAhB〉S = GAB =(Ji(x)j(x′) 0

0 ρi(x)j(x′)

)(6.41)

If

HAB =(Ki(x)j(x′) 0

0 Hi(x)j(x′)

)(6.42)

then the action (6.41) is exactly the sum of the bosonic field action (6.12) andthe fermionic field action (6.28). In general, HAB may have non vanishing offdiagonal terms which are responsible for a coupling between the fermionicand the bosonic fields.

Again we kept the spinor indices hidden, so that ψi(x) actually meantψiα(x). If, instead, we take ψi(x,ξ), then the discrete spinor components arise


from the expansion of ψi(x,ξ) ≡ ψi(x, ξ) in terms of the Grassmann coordi-nates ξ ≡ ξμ. Our superfield can then be written as

Ψ = φi(x)ki(x) + ψi(x,ξ)hi(x,ξ) (6.43)

But in such expression for a superfield there is an assymmetry between thebosonic and the fermionic part. A more symmetric expression is

Ψ = φi(x,ξ)ki(x,ξ) + ψi(x,ξ)hi(x,ξ) (6.44)

so that both parts contain the commuting coordinates x and the anticom-muting (Grassmann) coordinates ξ. Then both fields, the commuting φi(x,ξ)

and the anticommuting ψi(x,ξ), can form a representation for the symplec-tic basis vectors (qμ, qμ) → (xμ, ∂/∂xμ) and for the orthogonal basis vectors(θμ, θμ) → (ξμ, ∂/∂ξμ). The action is then that of Eq. (6.40) in which thematrices GAB and HAB are given by suitably generalized Eqs. (6.41) and(6.42), in which instead of Ki(x)j(x′) and Hi(x)j(x′) we have Ki(x,ξ)j(x′,ξ′) andHi(x,ξ)j(x′,ξ′), respectively.

But isn’t such a theory in conflict with the connection between spin andstatistics that comes from the requirement of microcausality? Not necessarily.Had we taken for the submatrix Ki(x,ξ)j(x′,ξ′) in Eq. (6.42) a “Dirac equationlike matrix” such as (6.30), then the action (6.40) would contain a part witha bosonic field described by the Dirac action. This would be problematic.But a scalar field like matrix (6.19) would pose no problem. The fact thatthe bosonic field depends also on Grassmann coordinates, ξμ, means thatwe have a number of bosonic fields, coming from the expansion of φi(xμ, ξμ)in terms of ξμ. Each field is in agreement with microcausality, i.e., theircommutators at different times all vanish outside the light cone.

Such ‘superfield’ (6.44) with the action (6.40) seems to be a natural gen-eralization of the point particle in superspace considered in Sec. 4. A deeperinvestigation of this topics is beyond the scope of the present paper.

6.5. Fock space states

We will now explicitly show how the basis vectors, hi(x,ξ) and ki(x), that span,respectively, an orthogonal and a symplectic phase space, behave as creationand annihilation operators of a quantum field theory.

6.5.1. Fermionic fields: Generators of orthogonal Clifford algebras. Let

Ψ = ψi(x,ξ)hi(x,ξ) = ψiα(x)hiα(x), i = 1, 2, α = 1, 2, ..., 2n, (6.45)

be a vector of the fermionic subspace of the total (super) phase space, wherethe basis vectors satisfy

hi(x,ξ) · hj(x′,ξ′) = δijδ(x)(x′)δ(ξ)(ξ′), (6.46)

orhiα(x) · hjβ(x′) = δijδ(x)(x′)δαβ . (6.47)

Introducing the Witt basis,

hα(x) = 1√2(h1α(x) + ih2α(x)), (6.48)


hα(x) = 1√2(h1α(x) − ih2α(x)), (6.49)

the anticommutation relations (6.46),(6.47) become the familiar relations forfermionic creation and annihilation operators:

hα(x) · hβ(x′) = δαβδ(x)(x′), (6.50)

hα(x) · hβ(x′) = 0, hα(x) · hβ(x′) = 0. (6.51)

A possible vacuum state is the product of all operators hα(x):

Ω =∏α,x

hα(x), (6.52)

hα(x)Ω = 0. (6.53)A basis of the Fock space is then

{hα1(x1)hα2(x2) ... hαr(xr)Ω} , r = 0, 1, 2, 3, .... (6.54)

For a fixed point x we have

{hα1(x)hα2(x) ... hαr(x)Ω}, r = 0, 1, 2, ..., D, (6.55)

or, more explicitly,

{Ω, hα1(x)Ω, hα1(x)hα2(x)Ω, ... , hα1(x)hα2(x)...hαD(x)Ω}. (6.56)

Here D = 2n is the number of the basis vectors hα(x), i.e., the creationoperators (6.48) that arise from expansion of the basis vectors h(x,ξ) in termsof n Grassmann coordinates ξ ≡ ξμ, μ = 1, 2, ...,n. We now also consider thecase in which the number n of anticommuting (Grassmann) coordinates isdifferent from the number n of the commuting coordinates, xμ, μ = 1, 2, ..., n.This is distinguished by using normal and bold symbols. In particular, D =16, if n = 4 (which was our choice), and D = 4, if n = 2 (which is the usualchoice). The dimension of the Fock space spanned over the basis (6.56) is 2D.In particular,

2D ={

216 if n = 424 if n = 2 (6.57)

Besides the choice of vacuum (6.52), there are other possible choices,analogous to those considered in Sec. 4.4, for example,

Ω =∏α,x

hα(x) , hα(x)Ω = 0. (6.58)

More generaly, we have

Ω =

⎛⎝ ∏

α∈R1, x

hα(x)

⎞⎠

⎛⎝ ∏

α∈R2, x

hα(x)

⎞⎠ . (6.59)

Here R = R1 ∪R2 is the set od indices α = 1, 2, ..., D, where R1 and R2 aresubsets of indices, e.g., R1 = {1, 4, 7, ..., D} and R2 = {2, 3, 5, 6, ...D − 1}.Altogether, with respect to all such arrangements of indices α, there are 2D

possible vacua, giving 2D×2D = 22D basis states. Since we consider operatorshα(x) hα(x) at a fixed point x, we can factor out from Ω the part due to the


product over x, and so those 22D states span a Clifford algebra Cl(2D). Inparticular, we have Cl(32), if n = 4, and Cl(8), if n = 2. To sum up, aClifford albegra Cl(2D) is generated at every point x by a set of 2D basisvectors, (hα(x), hα(x)), or equivalently, by (h(x,ξ), h(x,ξ)), which are fermioniccreation and annihilations operators.

If we do not factor out from Ω the part due to the product over pointsx ∈ R

n, then it turns out that there are many other possible definitions ofvacuum, such as

Ω =

⎛⎝ ∏

α, x∈R1

hα(x)

⎞⎠

⎛⎝ ∏

α, x∈R2

hα(x)

⎞⎠ , (6.60)

depending on a partition of Rn into two domains R1 and R2 so that Rn =

R1∪R2. Instead of the configuration space, we can take the momentum space,and consider, e.g., positive and negative momenta p. If Rn is the Minkowskispacetime, then we can have a vacuum of the form

Ω =

⎛⎝ ∏

α, p0>0,p

hα(p0,p)

⎞⎠

⎛⎝ ∏

α, p0<0,p

hα(p0,p)

⎞⎠ (6.61)

which is annihilated according to

hα(p0>0,p) Ω = 0 , hα(p0<0,p) Ω = 0, (6.62)

whereas one particle states are created according to

hα(p0>0,p) Ω , hα(p0<0,p) Ω. (6.63)

With respect to the above vacuum (6.61), one kind of particles are createdby positive energy unbarred operators hα(p0>0,p), whilst the other kind ofparticles are created by negative energy barred operators hα(p0<0,p). Thevacuum with reversed properties can also be defined, besides many otherpossible vacua. All those vacuum definitions participate in a description ofthe interactive processes of elementary particles. What we take into accountin our current quantum field theory calculations seem to be only a part of alarger theory that has been neglected. It could be that some of the difficul-ties (e.g., infinities) that we have encountered, are partly due to neglection ofsuch a larger theory. For instance, the vacuum (6.62) is considered by Jackiwet al. [62],[63] within the context of a 2-dimensional field theory with signa-ture (+−). In Refs. [64] it is shown how with such definition of vacuum wecan obtain vanishing zero point energy, and yet the Casimir and other sucheffects remain intact. This could be a resolution [64] of the problem of thehuge cosmological constant predicted by the ordinary quantum field theory.Another application is in string theory which, as shown in Ref. [65], can beformulated in non critical dimensions.

6.5.2. Bosonic fields: Generators of symplectic Clifford algebras. Let us nowconsider a vector

Φ = φi(x)ki(x) , i = 1, 2 (6.64)


of the bosonic subspace of the total (super) phase space, where the basisvectors satisfy relation (6.9) of a symplectic Clifford algebra. In the basis

k′1(x) ≡ k(x) = 1√2(k1(x) + k2(x)), (6.65)

k′2(x) ≡ k(x) = 1√2(k1(x) − k2(x)) (6.66)

relations (6.9) become

k(x) ∧ k(x′) = 0, k(x) ∧ k(x′) = 0, (6.67)

k(x) ∧ k(x′) = δ(x)(x′), (6.68)

which, apart from a factor 1/2 that enters definition of the wedge product, arethe commutation relations for bosonic creation and annihilation operators.

In the case of fermionic operators, a possible vacuum was defined asthe product of all annihilation operators. For boson operators such definitiondoes not work. At the moment it is not clear to me whether a bosonic vacuumcan be defined in terms of creation and annihilation operators. Formally, wedefine

k(x)Ω = 0, (6.69)

where Ω now denotes a bosonic vacuum. The basis

{k(x1)k(x2)...k(xr)} , r = 0, 1, 2, ... (6.70)

spans a Fock space, whose vectors are

ΦF =∞∑r=0

φ(x1)(x2)...(xr)k(x1)k(x2)...k(xr)Ω , (6.71)

where components φ(x1)(x2)...(xr) are symmetric in (x1)(x2)...(xr).On the other hand, a vector (6.64) can be generalized to an element of

a symplectic Clifford algebra:

ΦC =∞∑r=0

φi1(x1)i2(x2)...ir(xr)ki1(x1)ki2(x2)...kir(xr), (6.72)

which contain both kinds of operators, k′1(x) ≡ k(x) and k′2(x) ≡ k(x). Itremains to be explored whether the vectors ΦF of the form (6.71) belong toa subspace of the space whose vectors are ΦC of Eq. (6.72). This is equivalentto the question of whether Ω can be defined in terms of k(x).

7. Discussion

7.1. Prospects for unification

We started from the super phase space action (4.38) in which the numberof commuting variables za(τ) = (xμ(τ), pμ(τ)) is the same as the number ofanticommuting variables λa(τ) = (λμ(τ), λμ(τ)), μ = 1, 2, ..., n. This arises, ifwe consider superfields Za(τ, ζ) which depend on a commuting parameter τ ,and on an anticommuting parameter ζ. So we have Za(τ, ζ) = za(τ)+ζλa(τ).


After (first) quantization we arrived at the action (6.28) or (6.33) fora vector field Ψ = ψi(x,ξ)hi(x,ξ) = ψiα(x)hiα(x), α = 1, ..., 2n and i = 1, 2.The latter index denotes a field and its canonically conjugate field. The basisvectors hiα(x), satisfying relations (6.47), are generators of an infinite di-mensional Clifford algebra, and they have the role of quantized fields. If wetransform them into another basis according to (6.48), (6.49), then the newoperators hα(x), hα(x) satisfy the anticommutation relations (6.50), (6.51) forfermionic fields. If we start from 4-dimensional spacetime, then the index αassume 24 = 16 values. Therefore, the set of fields {hα(x), hα(x)} is biggerthan it is necessary for description of a spin 1

2 particle and its antiparticle,in which case four values of α are sufficient. We now have a possibility that{hα(x), hα(x)} describe electron, neutrino, and, e.g., the corresponding mirrorparticles, as shown in Ref. [28].

In order to describe other particles and gauge interactions of the stan-dard model, one has to extend the theory. One possibility [30, 31, 66] isto replace spacetime with Clifford space [13]–[15], which is a manifold of di-mension N = 2n, whose tangent space at any point X is a Clifford alge-bra Cl(n). Clifford space is a quenched configuration space associated withp-branes [67, 68, 66]. One can then proceed as we did in this paper, andarrive at the set of fermionic field operators {hα(X), hα(X)}, where α nowruns from 1 to 2N , because we replaced n-dimensional spacetime with N -dimensional Clifford space. The theory then becomes analogous to a unifica-tion in the presence of higher dimensions. Another possibility is to exploitthe 2D-dimensional Fock space spanned over the basis (6.56), and take intoaccount the fact that the latter Fock space is a left minimal ideal of a Cliffordalgebra Cl(2D), and is thus the space of spinors. Various approaches to theunification of fundamental particles and forces by Clifford algebras have beenexplored in Refs. [29]–[46], [28].

7.2. Prospects for quantum gravity

We have seen in Sec. 2.3 that the generators γa = (γμ, γμ), μ = 1, 2, ..., n,of an orthogonal Clifford algebra Cl(2n) can be rewritten in terms of newgenerators, θa = (θμ, θμ), which satisfy the fermionic anti commutation rela-tions (2.27). In Sec. 4.4. we then show that θμ, θμ act as fermionic creationand annihilation operators, from which we can build basis spinors, sA, of allminimal left ideals of Cl(2n). A generic element is Ψ = ψAsA, where thecoefficients ψA may depend on spacetime position xμ. An interesting objectto consider is

〈γμ〉1 = 〈Ψ‡(x)γμΨ(x)〉1 , (7.1)

i.e., the vector part of the expectation value of a vector γμ. Since Ψ is ageneric element of Clifford algebra, a “Clifford aggregate” or polyvector, theexpectation value is a linear superposition of vectors γμ′ :

〈γμ〉1 = eμν′

(x) γν′ . (7.2)


Here γν′ are orthogonal vectors, whilst 〈γμ〉1 need not be orthogonal, andeμ

ν′may serve the role of vielbein. There is a possibility that the vectors

〈γμ〉1 are tangent vectors to manifold with non vanishing curvature, so thattheir inner product

〈γμ〉1 · 〈γν〉1 = gμν(x) (7.3)

gives a metric that cannot be transformed into ημν at every point x. If it isindeed the case that the curvature can be different from zero, then curvedspace(time) can be generated from position dependent spinors. This remainsto be explored, and if it turns to be true, this will have implications forquantum gravity.

8. Conclusion

In this work we have pointed out how ‘quantization’ can be seen from yetanother perspective. We reformulated and generalized the theory of quan-tized fields. An action for a physical system, such as a point particle or afield, can be written in the phase space form, and it contains either the sym-plectic or the orthogonal form (or both). The corresponding basis vectorssatisfy either the fermionic anticommutation relations or the bosonic com-mutation relations. If we take a Hamiltonian that is quadratic in the phasespace variables, derive the classical equations of motion and then assumethat coordinates and momenta are undetermined, it turns out that the ba-sis vectors satisfy the Heisenberg equations of motion. Quantum mechanicaloperators are just the basis vectors included in the phase space action. Theycan be expressed as creation and annihilation operators acting on a vacuumwhich is the product of annihilation operators (in the fermionic case). In thefinite dimensional case this gives the Fock basis for spinors. If we considernot only one vacuum, but all possible vacua, then we obtain the Fock basisfor a Clifford algebra [26, 27, 28]. In infinite dimensional case, i.e., in the caseof fields, we obtain a more general Fock basis and many possible vacua thatgo beyond those usually considered in quantum field theories. It would beinteresting to explore whether such a generalized quantum field theory is freeof the difficulties, such as infinities and the cosmological constant problem.

As a particular model we considered a point particle described in termsof commuting and an equal number, n, of anticommuting (Grassmann) phasespace variables. The phase space action (4.38) contains a form that consistsof the latter variables and the corresponding basis vectors— the generatorsof an orthogonal, Cl(2n), and a symplectic Clifford algebra, ClS(2n). If westart from a 4-dimensional spacetime, i.e., if we take n = 4, then we obtainthe spinor states—created from the basis vectors—of sufficiently high dimen-sionality that they can be considered in our attempts for grand unification.Finally, we showed how the fact that the basis vectors on the one hand arequantum mechanical operators, and on the other hand they give metric, couldbe exploited in the development of quantum gravity.


Acknowledgment

This work is supported by the Slovenian Research Agency.

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Matej PavsicJozef Stefan InstituteJamova 39SI 1000 LjubljanaSloveniae-mail: [email protected]

Received: May 20, 2011.

Accepted: May 24, 2011.

a theory of quantized fields based on orthogonal and symplectic clifford algebras

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