JMathPhys_38_1347

Downloaded¬13¬Apr¬2 On quantum field theory with nonzero minimal uncertainties in positions and momenta Achim Kempfa) Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, United Kingdom ~Received 13 February 1996; accepted for publication 1 November 1996! We continue studies on quantum field theories on noncommutative geometric spaces, focusing on classes of noncommutative geometries which imply ultraviolet and infrared modifications in the form of nonzero minimal uncertainties in posi- tions and momenta. The case of the ultraviolet modified uncertainty relation which has appeared from string theory and quantum gravity is covered. The example of Euclidean f4-theory is studied in detail and in this example we can now show ultraviolet and infrared regularization of all graphs. © 1997 American Institute of Physics. @S0022-2488~97!01403-5# I. INTRODUCTION There has been considerable progress in several branches of the mathematics of noncommu- tative or ‘‘quantum’’ geometry which, in a broad sense, is the generalization of geometric con- cepts and tools to situations in which the algebra of functions on a manifold becomes noncom- mutative. The physical motivations range, e.g., from integrable models and generalized symmetry groups to studies on the algebraic structure of the Higgs sector in the standard model. Standard references are, e.g., 1–9. Here, we continue the approach of Refs. 10–16 in which is studied the quantum mechanics on certain ‘‘noncommutative geometries’’ where @xi ,xj#Þ0 and @pi ,pj#Þ0, ~1! and in particular where @xi ,p j#5i\~d i1a i jklxkxl1b i jklpkpl1 . . . !. ~2! A crucial feature of the generalized commutation relations, which we will discuss in Sec. II, is that for appropriate matrices a ,b P Mn4(C) one finds ordinary quantum mechanical behaviour at me- dium scales, while as a new effect at very small and very large scales there appear nonzero minimal uncertainties Dx0 ,Dp0 in positions and in momenta. The main part of the paper is Sec. III, where we proceed with the study of a previously suggested approach14 to the formulation of quantum field theories on such geometries. For the example of f4-theory, we can now explicitly show that minimal uncertainties in positions and momenta do have the power to regularize all graphs in the ultraviolet and the infrared. The underlying motivation is the idea that nonvanishing minimal uncertainties in positions and momenta could be effects caused by gravity, or string theory. The possible gravitational origins for modifications in the ultraviolet and in the infrared are to be considered separately. On the one hand, in order to resolve small distances, test particles need high energies. The latest at the Planck scale of about 10235 m the gravity effects of high energetic test particles must significantly disturb the spacetime structure which one tries to resolve. It has therefore long been suggested that there exists a finite limit to the possible resolution of distances. Probably the simplest ansatz for its quantum theoretical expression is that of a nonvanishing minimal uncer- a!Electronic-mail: a.kempf@damtp.cambridge.ac.uk 0022-2488/97/38(3)/1347/26/$10.00 1347J. Math. Phys. 38 (3), March 1997 © 1997 American Institute of Physics 011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions 1348 Achim Kempf: Quantum field theory with finite x,p-uncertainties Downloaded¬13¬Apr¬2 tainty in positions. This ansatz covers an ultraviolet behaviour which has been found in string theory, as well as in quantum gravity, arising from an effective uncertainty relation: Dx> \ Dp1const•Dp . ~3! References are, e.g., 17–23; a recent review is Ref. 24. On the other hand, minimal uncertainties in momentum, as an infrared effect, may arise from large scale gravity. The argument is related to the fact that on a general curved spacetime there is no notion of a plane wave, i.e., of exact localization in momentum space, see Refs. 14, 16. We remark that in the case of minimal uncertainties in positions only, examples are known of noncommutative geometries of the type of Eqs. ~1! and ~2! which preserve the Poincare´ symmetry, i.e., where the universal enveloping algebra of the Poincare´ Lie algebra is a *- sub algebra of the Heisenberg algebra, see Refs. 25, 26. Generally, however, we take the view that similarly to curved spaces which may preserve some of the flat space symmetries while breaking others, also noncommutative geometric spaces, as defined through commutation relations, may preserve some symmetries while breaking others. Here, we therefore study the general case, i.e., not assuming a specific symmetry, and allow- ing the existence both of minimal uncertainties in positions and in momenta. An alternative approach with a similar motivation, but based on the canonical formulation of quantum field theory, is Ref. 27. Other approaches to nonrelativistic quantum mechanics with generalized com- mutation relations, mostly motivated by quantum groups, and related studies, are, e.g., 28–44. II. QUANTUM MECHANICS WITH NONZERO MINIMAL UNCERTAINTIES A. Uncertainty relations We review and generalize the results of Refs. 10–13 on nonrelativistic quantum mechanics with nonzero minimal uncertainties in positions and momenta. Let A denote the associative Heisenberg algebra generated by elements xi ,pj that obey generalized commutation relations of the form of Eqs. ~1! and ~2!. The modified commutation relations are required to be consistent with the *-involution xi* 5 xi, pj* 5 pj, implying that a and b obeya i jkl* 5 a i j lk ,b i jkl* 5 b i j lk . The study of the uncertainty relations that belong to the Heisenberg algebra A yields infor- mation that holds independently of the choice of representation. Let us therefore assume the xi ,pj to be represented as symmetric operators obeying the new commutation relations on some dense domain D,H in a Hilbert space H. On this space D of physical states one derives uncertainty relations of the form DADB>1/2u^@A ,B#&u ~4! so that, e.g., @xi ,xj#Þ0, yields DxiDx j>0. Their noncommutativity implies that the xi ~as well as the pı! can no longer be simultaneously diagonalised. Because of the modified commutation relations Eqs. ~2! and the corresponding uncertainty relations there can appear the even more drastic effect that the xi ~as well as the pj! may also not be diagonalisable separately. Instead there then exist nonzero minimal uncertainties in positions and momenta. The mechanism can be seen also in one dimension, to which case we will restrict ourselves until Sec. II E. We consider Eq. ~2! with a,b.0 and ab,1/\2: @x,p#5i\~11ax21bp2!. ~5! J. Math. Phys., Vol. 38, No. 3, March 1997 011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions 1349Achim Kempf: Quantum field theory with finite x,p-uncertainties Downloaded¬13¬Apr¬2 For fixed but sufficiently small a and b one finds ordinary quantum mechanical behaviour at medium scales while, e.g., the term proportional to b contributes for large ^p2&5^p&21(Dp)2, i.e., in the ultraviolet. Similarly the term proportional to a leads to an infrared effect. The uncertainty relation to Eq. ~5! is: DxDp> \ 2 ~11a~Dx ! 21a^x&21b~Dp !21b^p&2!. ~6! It implies nonzero minimal uncertainties in x- as well as in p- measurements. This can be seen as follows: As, e.g., Dx gets smaller, Dp must increase so that the product DxDp of the LHS remains larger than the RHS. In usual quantum mechanics this is always possible, i.e., Dx can be made arbitrarily small. However, in the generalized case, for a,b.0 there is a positive (Dp)2 term on the RHS which eventually grows faster with Dp than the LHS. Thus Dx can no longer become arbitrarily small. The minimal uncertainty in x depends on the expectation value in position and momentum via k:5a^x&21b^p&2 ~7! and is explicitly: Dx05A~11k !b\212ab\2 . ~8! Analogously one obtains the smallest uncertainty in momentum Dp05A~11k !a\212ab\2 ~9! with the absolutely smallest uncertainties obtained for k50. Note that if there was, e.g., an x- eigenstate uc&PD with x.uc&5luc& it would have no uncer- tainty in position ~we always assume states uc& to be normalized!: ~Dx ! uc& 2 5^cu~x2^cuxuc&!2uc&50 ~10! which would be a contradiction. There are thus no physical states uc&PD which are eigenstates of x or p. Thus for any physical domain D , i.e., for all *-representations of the commutation relations, there are no physical states in the ‘‘minimal uncertainty gap:’’ '” uc&PD:0<~Dx ! uc&,Dx0 , ~11! '” uc&PD:0<~Dp ! uc&,Dp0 . ~12! Crucially, unlike on ordinary geometry, there do not exist sequences $ucn&% of physical states which would approximate point localizations in position or momentum space, i.e., for which the uncertainty would decrease to zero: '” ucn&PD: lim n!` ~Dx ! ucn&50 or lim n!` ~Dp ! ucn&50. ~13! Heisenberg algebras A with these generalized canonical commutation relations therefore no longer have spectral representations on wave functions ^x uc& or ^p uc&. J. Math. Phys., Vol. 38, No. 3, March 1997 011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions 1350 Achim Kempf: Quantum field theory with finite x,p-uncertainties Downloaded¬13¬Apr¬2 B. Bargmann Fock representation For practical calculations and for detailed studies of the functional analysis a Hilbert space representation of the generalized Heisenberg algebra is needed. We generalize the Bargmann Fock representation. In ordinary quantum mechanics the Bargmann Fock representation is unitarily equivalent to the position and the momentum representation, being the spectral representation of the operator h¯]h¯PA,where h¯:5 1 2L x2 i 2K p and ]h¯ :5 1 2L x1 i 2K p. ~14! Here, L and K are length and momentum scales, related by LK5\/2. Thus h¯ and ]h¯ obey ]h¯h¯ 2 h¯]h¯51, which is of the form of a Leibniz rule and justifies the notation. One readily finds the countable set of eigenvectors h¯]h¯uh¯n&5nuh¯n& with n50,1,2,••• . With the definitions uah¯n1bh¯m&:5uah¯n&1ubh¯m& and auh¯n&:5uah¯n& arbitrary states uc& are written as polynomials or power series uc&5U( r50 ` cr h¯r Ar!L 5uc~h¯!& ~15! on which x and p are represented in terms of multiplication and differentiation operators x5L~ hˆ1]h¯!, p5iK~h¯2]h¯!. ~16! The well known formula for the scalar product of states is ^cuf&5 1 2pi E dn dh¯ c~h¯!e2h¯ hf~h¯!. ~17! Here, the c(h¯) and f(h¯) on the RHS are to be read as polynomials or power series in ordinary complex variables rather than as elements of A. A key observation for the generalization of the Bargmann Fock representation is that the scalar product can be expressed without relying on complex integration:10 ^cuf&5c~h¯!e]h]h¯f~h¯!uh505h¯ . ~18! The exponential is defined through its power series, i.e., e]h]h¯ 5 (r50 ` (]h]h¯)/(r!), where the derivatives ]h¯ act from the left while the derivatives ]h act from the right. The evaluation proce- dure is to carry out the differentiations and then to set h and h¯ equal to zero. The remaining number is the value of the scalar product. This can be done purely algebraically by using the Leibniz rule ]h¯h¯ 2 h¯]h¯ 5 1 and its complex conjugate h]h2]hh51. For example, ]h¯h¯ 25]h¯h¯h¯5~h¯]h¯11 !h¯5h¯]h¯h¯1h¯5h¯~h¯]h¯11 !1h¯5h¯h¯]h¯1h¯1h¯52h¯ and h2]h5h~]hh11 !5•••52h . J. Math. Phys., Vol. 38, No. 3, March 1997 011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions 1351Achim Kempf: Quantum field theory with finite x,p-uncertainties Downloaded¬13¬Apr¬2 Thus, e.g., ^h¯2u213h¯2&5h2e]h]h¯~213h¯2!uh505h¯5h2( r50 ` ]h]h¯ r! ~213h ¯ 2!uh505h¯ 53h2 ]h 2 ]h¯ 2 2 h ¯ 2uh505h¯56. Since the scalar product formula Eq. ~17! relies on conventional commutative integration over the complex plane, it cannot be used in the generalized case where, e.g., in n dimensions the h¯i will be noncommutative. It is, however, possible to use a generalization @Eq. ~24!# of Eq. ~18! ~which can also be applied in the fermionic case instead of using Berezin integration10!. Also in one dimension it allows to construct a Bargmann Fock Hilbert space representation for Eq. ~5!. To this end we rewrite Eq. ~5! in the form @x,p#5i\1i\~q221 ! S x24L2 1 p 2 4K2D , ~19! where the parameter q>1 measures the deviation from the ordinary commutation relations. The length and momentum scales are related by LK5\(q211)/4. We can now again represent x and p as the usual linear combinations @Eq. ~16!# of generators h¯ and ]h¯ . A complete generalized Bargmann Fock calculus is defined as the complex associative algebra B with the commutation relations ]h¯h¯2q2h¯]h¯51, h]h2q2]hh51, ~20! h¯]h2q2]hh¯50, ]h¯h2q2h]h¯50, ~21! hh¯2q2h¯h50, ]h]h¯2q2]h¯]h50. ~22! A short calculation shows that the commutation relation Eq. ~19! in fact uniquely translates into the commutation relations Eqs. ~20! through Eq. ~16!, see Ref. 13. On the other hand, the com- mutation relations Eqs. ~21! and ~22! are nonunique and could also be chosen commutative. Our choice is the special case of the choice made for the n dimensional case in Ref. 10 under the requirements of a quantum group module algebra structure, invariance of the Poincare´ series and simple form of the scalar product formula. These requirements are here not physically relevant, but it is convenient to use the formulas already obtained for this case. Generally, other choices for the commutation relations between the barred and the unbarred generators are possible and lead to, respectively, more or less simple to evaluate formulations of the scalar product. These represen- tation specific choices do not of course affect the physical content of the theory, such as the uncertainty relations, transition amplitudes or expectation values. The Heisenberg algebra A is now represented on the domain D of polynomials in h¯ D:5$uc&uc~h¯!5polynomial~h¯!% ~23! with the action of x and p given by Eq. ~16!, where the differentiations are to be evaluated algebraically using the generalized Leibniz rule given in Eqs. ~20!. As is the case on ordinary geometry, the operators h¯ and ]h¯ are mutually adjoint with respect to the unique and positive definite scalar product, which now takes the form: ^cuf&5c~h¯!e1/q ]h]h¯f~h¯!uh505h¯ . ~24! J. Math. Phys., Vol. 38, No. 3, March 1997 011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions 1352 Achim Kempf: Quantum field theory with finite x,p-uncertainties Downloaded¬13¬Apr¬2 The q-exponential is defined through e1/q ]h]h¯5( r50 ` ~]h i]h¯i! r @r#1/q! , ~25! where the derivatives ]h act from the right and where @r#c :511c21c41•••1c2~r21 !5 c2r21 c221 and @r#c!:51•@2#c•@3#c• ••• •@r#c . The evaluation procedure is again to algebraically carry out the differentiations, now using Eqs. ~20!–~22! and then to set h and h¯ equal to zero. The remaining number is the value of the scalar product. The functional analysis of the position and momentum operators is as follows: We denote by H the Hilbert space obtained by completion with respect to the norm induced by the scalar product. A Hilbert basis is given by the orthonormal family $~@r#q! !21/2uh¯r&ur50,1,2,.. .%. ~26! The domain D,H , which is dense in H , is a physical domain, i.e., on it the x and p are represented as symmetric operators obeying the commutation relation Eq. ~19!. In fact D is also analytic since x.D,D and p.D,D , i.e., D is a *-A module. The x and p are no longer essen- tially self-adjoint. Their adjoints x* and p* are closed but nonsymmetric. The x** and p** are closed and symmetric. Their deficiency subspaces are of finite ~nonzero! and equal dimension so that there are continuous families of self-adjoint extensions in H . Crucially, however, because of the minimal uncertainties in positions and momenta, neither x nor p have self-adjoint extensions neither in D nor in any other physical domain, i.e., not in any other *-representation of the commutation relations. For the details and proofs see Ref. 13. One arrives at the following picture: While in classical mechanics the states can have exact positions and momenta, in quantum mechanics there is the uncertainty relation that does not allow x and p to have common eigen- vectors. Nevertheless x and p separately do have ‘‘eigenvectors,’’ though non-normalisable ones. The spectrum is continuous, namely, the configuration or momentum space. The position and momentum operators are essentially self-adjoint. Our generalization of the Heisenberg algebra has further consequences for the observables x and p: It is not only that the x and p have no common eigenstates. The uncertainty relation now implies that they do not have any eigenvectors in the representation of the Heisenberg algebra. Although x and p separately do have self-adjoint exten- sions, they do not have self-adjoint extensions on any physical domain i.e., not on any *-representation of both x and p. This means the nonexistence of absolute precision in position or momentum measurements. Instead there are absolutely minimal uncertainties in these measure- ments which are, in terms of the new variables of Eq. ~19!: Dx05LA12q22, Dp05KA12q22. ~27! Recall that due to Eq. ~10! the non-self-adjointness and nondiagonalisability of x and p is neces- sary to allow for the physical description of minimal uncertainties. Note that, on the other hand, the fact that x and p still have the slightly weaker property of being symmetric is sufficient to guarantee that all physical expectation values are real. J. Math. Phys., Vol. 38, No. 3, March 1997 011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions 1353Achim Kempf: Quantum field theory with finite x,p-uncertainties Downloaded¬13¬Apr¬2 C. Maximal localization states Generally, all information on positions and momenta is encoded in the matrix elements of the position and momentum operators, and matrix elements can of course be calculated in any basis. In the Bargmann Fock basis matrix elements, e.g., of the position operators are calculated as ^cuxuf&5c~h¯!e1/q ]h]h¯L~h¯1]h¯!f~h¯!u0 . ~28! Ordinarily, information on position or momentum can conveniently be obtained by projection onto position or momentum eigenstates ^x uc& or ^p uc&, i.e., by using a position or momentum repre- sentation. That there are now no more physical x- or p-eigenstates, can also be seen directly in the Bargmann Fock representation. We consider, e.g., the eigenvalue problem for x x.ucl&5lucl&, i.e., L~h¯1]h¯!cl~h¯!5lcl~h¯! ~29! which yields a recursion formula for the coefficients of the expansion: cl~h¯!5( r50 ` cl ,rh¯ r . ~30! In ordinary quantum mechanics the solution is a Dirac d ‘‘function,’’ transformed into Bargmann Fock space, @i.e., Eq. ~102! with l instead of x0#. In the generalized setting, it is interesting to see the effect of the appearance of the minimal uncertainty ‘‘gap.’’ The ~no longer generally mutually orthogonal! solutions (r50` cl ,rh¯ r to Eq. ~29! have van- ishing uncertainty in positions but they are not contained in the domain of p ~this would of course contradict the uncertainty relation! and they are therefore not physical states. However, every polynomial approximation to the power series is contained in the physical domain D , i.e., (r50 n cl ,rh¯ rPD for arbitrary finite n . Thus each (r50n cl ,rh¯ r has an x- uncertainty which is in fact larger than Dx0 . For details and a graph of their scalar product see Ref. 13. Let us now consider the physical states ufj ,pmlx&,ufj ,pmlp& which have the maximal localization in x or p for given expectation values j,p in positions