Covariant Geometric Prequantization of Fields

2 Pages·0.09 MB·English

by  I. V. Kanatchikov

COVARIANT GEOMETRIC PREQUANTIZATION OF FIELDS
IGOR V.KANATCHIKOV
Institute of Theoretical Physics
Friedrich Schiller University, 07743 Jena, Germany
Tallinn Technical University, 19086 Tallinn, Estonia
1 AgeometricprequantizationformulaforthePoisson-Gerstenhaberbracketofforms
0 foundwithintheDeDonder-WeylHamiltonianformalismearlierispresented.The
0 relatedaspects ofcovariantgeometricquantization offieldtheories aresketched.
2

n Theuseofthemethodsofgeometricquantization1infieldtheoryisseverelylim-
a
itedbytheinfinitedimensionalityofthestandardfieldtheoreticcanonicalHamilto-
J
nianformalismbecauseofthedifficultiesofsubstantiationofformalgeneralizations
4
of geometric constructions to infinite dimensions.

2 WesuggestanapproachbasedontheDeDonder-Weyl(DW)Hamiltonianform
v of field equations
8 ∂ φa(x)=∂H/∂pµ, ∂ pµ(x)=∂H/∂φa, (1)
µ a µ a
3
where pµ := ∂L/∂φa , H(φa,pµ,xν) := φa pµ −L, L(φa,φa ,xν) is a Lagrangian
0 a ,µ a ,µ a ,µ
2 density, which incorporates the field dynamics into a finite dimensional polymo-
1 mentum phase space: (pµ,φa,xµ) and requires no space+time splitting. The Pois-
a
0 son bracket for the above DW formulation is defined on differential forms which
0 represent dynamical variables2,3. The basic structure is the polysymplectic form
/
c Ω := dφa ∧dpµ∧ω (ω := ∂ (dx1 ∧...∧dxn)) which maps (“horizontal”) f-
a µ µ µ
q
formsF to (“vertical”)multivectorsX ofdegree(n−f): X Ω=dF. Themap
- F F
r exists for the so-calledHamiltonian forms the space ofwhichis closedwithrespect
g to the co-exterior product3: F•G:=∗−1(∗F∧∗G). The bracket
:
v {[F,G]}:=(−)n−f£X (G), (2)
i F
X where £XF :=[XF,d]=XF ◦d−(−)n−fd◦XF, equips the space of Hamiltonian
r forms with a Gerstenhaber algebra structure2,3,4:
a
{[F,G]}=−(−1)g1g2{[G,F]},
(−1)g1g3{[F,{[G,K]}]}+(−1)g1g2{[G,{[K,F]}]}+(−1)g2g3{[K,{[F,G]}]}=0, (3)
{[F,G•K]}={[F,G]}•K+(−1)g1(g2+1)G•{[F,K]},
where g =n−f −1, g =n−g−1, g =n−k−1.
1 2 3
How to quantize fields using the above generalizationof Poissonbrackets? Our
recent workon “precanonicalquantization”5−12 based on heuristic quantizationof
asmallsubalgebraof(3)leadstoageneralizationofquantumtheoreticformalismin
which the polymomenta operators are pˆµ = −i¯hκγµ ∂ and the covariant “multi-
a ∂φa
temporal” analogue of the Schr¨odinger equation for Ψ=Ψ(φa,xµ) reads
i¯hκγµ∂ Ψ=HΨ, (4)
µ b
where the constant κ∼ 1 is of the ultra-violet cutoff scale and, for the
[length](n−1)
scalarfield, H =−1¯h2κ2∂2 +V(φ). Itisnotable thatEq.(4)enablesusto derive
b 2 φφ
thestandardfunctionaldifferentialSchr¨odingerequationinquantumfieldtheory15.
submitted to: Proc. Ninth Marcel Grossmann Meeting, Roma (Italy), July 2-8, 2000 1
The main result of the present contribution is a generalizationof a cornerstone
ofgeometricquantization,theprequantizationformula,tothepresentframework13:
the operator
Fˆ =i¯h£ +(X Θ)•+F•, (5)
XF F
with dΘ:=Ω, is a prequantization of a Hamiltonian form F in the sense that
[Fˆ,Fˆ]=−i¯h{[F ,F ]}, (6)
1 2 d1 2
where [A,B]=A◦B−(−1)degAdegBB◦A is a gradedcommutator. The operator
(5) is nonhomogeneous: the degree of the first term is (n−f −1) while the degree
oftheothertwois(n−f). Therefore,thefirsttwotermscanbeviewedasacovari-
antderivative∇ correspondingto a superconnection14; thenthe polysymplectic
XF
form can be viewed as the curvature of the superconnection. Prequantum opera-
tors(5)canactonprequantumHilbertspaceofnonhomogeneousdifferentialforms
Ψ(φa,pνa,xν)=Pnp=0ψµ1...µpdxµ1∧...∧dxµp with the scalar product hΨ,Ψi(x)=
Pn R(ψ ψµ1...µp)Vol,where Vol:=Qm dφa∧dp1∧dp2∧...∧dpn (no sum-
p=0 µ1...µp a=1 a a a
mation over index a here!). However, it is more suitable to Cliffordize the above
expressions according to the rule ω • = κ−1γ . Then the Cliffordized prequan-
µ µ
tum operators (5) can act on spinor wave functions Ψ(φa,pν,xν) with the positive
a
definite scalar product hΨ,Ψi = R RσΨγµΨ Vol∧ωµ, where σ is a space-like hy-
persurface. Using the “vertical polarization” with Ψ = Ψ(φa,xν) one can derive
from (5) the operators already known in precanonicalquantization5−12. Thus, the
structures ofDW theory2,3 naturally leadto the notions ofClifford/spinorbundles
and superconnections14 as a framework of generalizing the techniques of geomet-
ric quantization1 to field theory, requiring neither a space+time splitting nor an
infinite dimensional geometry.
References
1. A.Echeverria-Enriqueze.a., ExtractaMath.13(1998)135,math-ph/9904008.
2. I.V. Kanatchikov,Rep. Math. Phys. 41 (1998) 49, hep-th/9709229.
3. I.V. Kanatchikov,Rep. Math. Phys. 40 (1997) 225, hep-th/9710069.
4. C. Paufler, Rep. Math. Phys. 47 (2001) to appear, math-ph/0002032.
5. I.V. Kanatchikov, Int. J. Theor. Phys. 37 (1998) 333, quant-ph/9712058.
6. I.V. Kanatchikov, Rep. Math. Phys. 43 (1999) 157, hep-th/9810165.
7. I.V. Kanatchikov,AIP Conf. Proc. v. 453 (1998) 356, hep-th/9811016.
8. I.V. Kanatchikov, in: Current Topics in Mathematical Cosmology, eds. M.
RainerandH.-J.Schmidt(WorldScientific, Singapore,1998)gr-qc/9810076.
9. I.V. Kanatchikov,Jena preprint FSU TPI 01/99,gr-qc/9909032.
10. I.V. Kanatchikov,Quantization of gravity: yet another way, gr-qc/9912094.
11. I.V. Kanatchikov,Int. J. Theor. Phys. 40 (2001) to appear, gr-qc/0012074.
12. I.V.Kanatchikov,Nucl.Phys.B Proc.Suppl. 88(2000)326,gr-qc/0004066.
13. I.V. Kanatchikov, Geometric prequantization of Hamiltonian forms in field
theory. in preparation.
14. D. Quillen, Topology 24 (1985) 89; ibid. 25 (1986) 85.
15. I.V. Kanatchikov, Precanonical quantization and the Schr¨odinger wave func-
tional, Jena preprint FSU TPI 08/00, hep-th/0012084
submitted to: Proc. Ninth Marcel Grossmann Meeting, Roma (Italy), July 2-8, 2000 2