Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS Clifford Algebras and Their Decomposition into Conugate Fermionic Heisenberg Algebras To cite this article: Sultan Catto et al 016 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - The remarkable algebra so*(n), its representations, its Clifford algebra and potential applications A O Barut and A J Bracken - ALGEBRAS THAT ARE DEFORMABLE INTO FREE ALGEBRAS A A Boyarkin - Some examples of finitely generated algebras V T Markov This content was downloaded from IP address on 0/1/017 at 17:3

2 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 Clifford Algebras and Their Decomposition into Conugate Fermionic Heisenberg Algebras Sultan Catto 1,,, Yasemin Gürcan 3, Amish Khalfan 4,, Levent Kurt 5 and V. Kato La 6 1 Physics Department, The Graduate School, City University of New York, New York, NY Theoretical Physics Group, Rockefeller University, 130 York Avenue, New York, NY , 5 Department of Science, Borough of Manhattan CC, The City University of NY, New York, NY Physics Department, LaGuardia CC, The City University of New York, LIC, NY Columbia University, New York, NY 1007 Work supported in part by the PSC-CUNY Research Awards, and DOE contracts No. DE-AC-076 ER and 03075; and NSF Grant No. DMS Work supported in part by the PSC-CUNY Research Award No: Abstract. We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them. 1. Clifford Algebras These are associative hermitian matrix algebras [1] with basis γ α ( α = 1,, n + 1) that obey γ α γ β = 1 {γ α, γ β } = 1 (γ αγ β + γ β γ α ) = δ αβ I (1) where I is a unit matrix of the same dimension as γ α. We have γ α = γ α, and γ α (γ β γ ρ ) = (γ α γ β )γ ρ () Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, ournal citation and DOI. Published under licence by Ltd 1

3 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 The basis elements γ α are linearly independent. The Clifford relation means that the square of each γ α is unity and two different γ α anticommute. (a) Suppose there are n such elements γ α. Define Then Also γ n+1 = ɛγ 1 γ γ n 1 γ n (3) {γ n+1, γ i } = 0 (i = 1,, n) (4) γ n+1 = ɛ ( 1) n (5) so that if ɛ = ( 1) n we have γ n+1 = 1. Thus the basis is generated by n elements γ 1,, γ n. (b) Construction of the algebra (Clifford, Dirac, Jordan and Wigner): Let n = 1. The answer is given by the hermitian traceless Pauli matrices τ 1, τ, τ 3. In this case τ 3 = iτ 1 τ. we have τ i τ + τ τ i = δ i I (6) where I is a unit matrix. The τ i act on complex -dimensional vector space U. We can also write τ = ie where e are the 3 quaternion units. Let n =. We take two commuting sets of Pauli operators σ (1) i and σ () acting in the vector space U U. We have seen that are 4 4 matrices with the properties σ (1) i = τ i I and σ () = I τ i (7) We define [σ (1) i, σ () ] = 0 (8) γ 1 = σ () 1, γ 3 = σ () γ = σ () 3 σ(1) 1, γ 4 = σ () 3 σ(1) (9) Then we find γ 5 = σ (1) 3 σ() 3 and {γ α, γ β } = δ αβ (10) Dirac s notation: σ (1) = σ, σ () i = ρ i. Then γ 1 = ρ 1, γ 3 = ρ, γ = ρ 3 γ 1, γ 4 = ρ 3 σ, γ 5 = ρ 3 σ 3. Next we look at its generalization. For arbitrary n we construct n + 1 different n n matrices that act on the vector space

4 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 U } U {{ U } n factors of dimension n in terms of n commuting sets of Pauli matrices (11) σ (n) = τ I I I Then we construct the following n + 1 hermitian matrices: σ (n 1) = I τ I I σ (1) = I I τ (1) They satisfy γ 1 = σ (n) 1, γ n+1 = σ (n) γ = σ (n) 3 σ(n 1) 1, γ n+ = σ (n) 3 σ(n 1) γ 3 = σ (n) 3 σ(n 1) 3 σ (n ) 1, γ n+3 = σ (n) 3 σ(n 1) 3 σ (n ) γ n+1 = σ (n) 3 σ(n 1) 3 σ () 3 σ(1) 3 (13) γ α γ β + γ β γ α = δ αβ I (α, β = 1,, n + 1) (14) (c) General Representation of the Clifford Algebra: The automorphism group of the Clifford algebra is the unitary group U( n ). Indeed, if then γ α = Uγ α U, (UU = I) (15) γ αγ β + γ β γ α = Uγ α U Uγ β U + Uγ β U Uγ α U = U(δ αβ I)U = δ αβ I (16) For example the following different representations of the Dirac algebra: I : γ 1 = ρ σ 1, γ = ρ σ, γ 3 = ρ σ 3, γ 4 = ρ 1, γ 5 = ρ 3 Weyl rep. (17) II : γ 1 = ρ σ 1, γ = ρ σ, γ 3 = ρ σ 3, γ 4 = ρ 3, γ 5 = ρ 1 Dirac rep. (18) 3

5 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 III : γ 1 = σ 1, γ = ρ σ, γ 3 = σ 3, γ 4 = σ ρ 1, γ 5 = σ ρ 3 Maorana rep. (19) are unitarily equivalent to the standard representation given above. (d) Reality properties in the standard representation: From the construction given above n + 1 elements of the basis (γ 1,, γ n and γ n+1 ) are real and symmetric while the remaining n elements (γ n+1,, γ n ) are imaginary and antisymmetric. (e) Generation of the n n matrix algebra from the Clifford algebra. γ α and I close under the symmetric Jordan product. They do not close under the matrix product since for instance where γ α γ β = 1 {γ αγ β } + 1 [γ α, γ β ] = 1 δ αβi + iσ αβ (0) σ αβ = 1 i [γ α, γ β ] (1) (called the spin matrices) are hermitian and linearly independent of γ α. Similarly γ α γ β γ ρ will contain a part independent of γ α and σ βρ up to products of γ α of degree n. The product of degree n + 1, namely γ 1 γ γ n γ n+1 () is proportional to the unit matrix as we have already seen. Hence a linear combination of all the powers of γ α from 0 to n span the whole n n matrix algebra. In the case of the Dirac algebra we have I, γ α and σ αβ respectively 1, 5, and 5 4 = 10 in number spanning the 16 independent elements of the 4 4 matrix algebra. (f) Let a = a α γ α. We have a = a α a α I. Let us now show an application: Take 4 of the γ α for n = : γ µ (µ = 1,, 3, 4). Let = γ µ x µ. Let ψ = mψ where ψ is a 4-column. Then ψ = µ µ ψ = ψ = m ψ (3) This is the relativistic wave equation for ψ. ψ = mψ is the Dirac equation. (g) Rotation of n + 1 dimensional vectors. 4

6 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 Consider an infinitesimal transformation of the vector a α This transformation is a rotation if a α = a α + δa α (4) This entails a α δa α = 0. Hence if a αa α = (a α + δa α )(a α + δa α ) = a α a α (5) we must have Now let a = γ ρ a ρ, δa = γ ρ δa ρ (6) {a, δa} = 0 (7) δa = [ i 4 σ αβω αβ, a] = [ i ω, a] (8) where ω = 1 σ αβω αβ. We have Hence {a, δa} = i (a[ω, a] + [ω, a]a) = i (aωa a ω + ωa aωa) = 0 (9) a a + i [ω, a] (30) represents a rotation and 1 σ αβ are called spin generators. In component form we write In the case of the Dirac algebra (n = ) a α = a α + ω αβ a β (31) 1 σ αβ (α, β = 1,, 5) (3) generate 5 dimensional rotations, while 1 σ µν (µ, ν = 1,, 4) (33) generate 4 dimensional rotations. When ω 3, ω 31, ω 1 are real and ω 41, ω 4, ω 43 are imaginary δa = [ i σ µνω µν, a] (34) for the vector a with a 1, a, a 3 real and a 4 imaginary represents a Lorentz transformation in 3 space and 1 time directions (the Minkowski space-time). Under such a Lorentz transformation while + [ i ω, ] (35) 5

7 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 ψ ψ + i ωψ (36) Thus transforms like a vector while ψ transforms like a spinor. Under such a Lorentz transformation the bilinear forms (where ψ = ψ γ 4 ) remains invariant, while transform like vectors.. Grassmann Algebra as Subalgebra of Clifford Algebra ψψ and i ψγ 5 ψ (37) ψγ µ ψ and i ψγ 5 γ µ ψ (38) Grassmann algebra and its conugate Grassmann algebra The Grassmann algebra is the algebra of associative anticommuting numbers. It is of order n if we can choose a basis b ( = 1,, n) such that and b i (b b k ) = (b i b )b k (39) {b i b } = b i b + b b i = 0, (or b i b = 0) (40) The associativity relation is automatically satisfied if b i is chosen to be matrices and they are multiplied with the rules of matrix multiplication. A standard set satisfying the anticommutativity relation can be obtained from the Clifford algebra of order n + 1 by choosing b = 1 (γ + iγ n+ ), ( = 1,, n) (41) We then verify that b = 0, and b b k = b k b for k. In the standard representation γ ( n) are real while γ n+ are imaginary so that b are real non-hermitian n n matrices. Another set of Grassmann numbers b of the same order can be obtained from b by a unitary transformation b = Ub U (4) with UU = I, where I is the n n unit matrix, or by the transformation to the conugate set which are also real and obey Let us now give examples: For n = 1 we have b = 1 ( (σ iσ ) = 0 b = 1 ( (σ 0 1 iσ ) = 1 b b = 1 (γ iγ n+ ) (43) b b k + b k b = 0 (44) ) 1 0 ) 0 0 (45) (46) 6

8 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 b = 0, b = 0 (47) b = UbU, and b = Ub U (48) are not real in general. For n = (Dirac algebra) we can construct a Grassmann algebra of order and its conugate, briefly: b 1 = ρ 1 + iρ b = ρ 3 σ 1 + iσ (49) (50) b 1 = b = 0, b1b = b b 1 (51) b 1 = ρ 1 iρ b = ρ σ 1 iσ 3 (5) (53) b 1 = b = 0, b 1 b = b b 1 (54) 3. Fermionic Heisenberg Algebra If the Grassmann algebra b and its conugate b are taken together we obtain b b k + b k b = 1 4 {γ + iγ n+, γ k iγ n+k } = δ k I (55) These relations are also invariant under the unitary transformation b Ub U. Thus from b and b k we obtain the Fermionic Heisenberg algebra {b, b k } = 0, {b, b k } = 0, {b, b k } = δ ki (56) These have the same structure as Poisson bracket relations among canonical variables (coordinates and momenta) in classical mechanics. The difference is that the Poisson bracket is anticommutative while the bracket used for the Fermionic Heisenberg algebra is commutative since it is an anticommutator. In the bosonic Heisenberg algebra commutator brackets with same formal properties as Poisson brackets are used. The n generators γ 1,, γ n of the Clifford algebra of order n + 1 can be recovered from the two conugate Grassmann algebras through γ = b + b, γ n+ = i(b b n+ ) (57) The last element γ n+1 of the Clifford basis can be expressed by means of b, b through the relation Now γ 1 γ n+1 γ γ n+ γ n γ n γ n+1 = i n I (58) γ γ n+ = i(i b b ) (59) 7

9 International Conference on Quantum Science and Applications (ICQSA-016) Journal of Physics: Conference Series 766 (016) doi: / /766/1/01001 so that 4. Further Remarks γ n+1 = n (I b b ) (60) =1 The present authors earlier developed octonion and split octonion [], [3] algebras that led into algebraic foundations of color symmetries in nature, into quark-antidiquark symmetries [4], [5], exotics and pentaquark formations. The formulations we put forth in this paper also have deeper applications in hadronic physics and in astrophysics. Further applications into Fueter analyticity, quaternion analyticity, twistor formalism, hyperbolic extensions of exceptional groups, M-theory and Leech lattices will be published in our forthcoming papers. References [1] S. Catto, Y. Gürcan, A. Khalfan and L. Kurt, Journal of Physics,411 (013) [] S. Catto, Y. Gürcan, A. Khalfan and L. Kurt, Journal of Physics,563 (014) [3] S. Catto, Y. Gürcan, A. Khalfan and L. Kurt, Journal of Physics,670 (016) [4] S. Catto and F. Gürsey, Nuovo Cimento A 86 (1985) 01 [5] S. Catto and F. Gürsey, Nuovo Cimento A99 (1988) 685 Acknowledgments: One of us (SC) would like to thank professor Abdullah Algın for the invited talk given at ICQSA-016 meeting in Eskişehir, Turkey, and for his kind hospitality. He is also grateful to late professor Feza Gürsey for useful discussions on many of the ideas presented in this coference. We would like to thank Professors Vladimir Akulov, Itzhak Bars and Francesco Iachello for many enlightening conversations. 8