JMathPhys_37_2121

Maximal localization in the presence of minimal uncertainties in positions and in momenta Haye Hinrichsena) Department of Physics of Complex Systems, Weizmann Institute, Rehovot 76100, Israel Achim Kempfb) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, United Kingdom ~Received 27 October 1995; accepted for publication 2 January 1996! Small corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed pro- vide natural cutoffs in quantum field theory. The corresponding underlying quan- tum theoretical framework includes small ‘‘noncommutative geometric’’ correc- tions to the canonical commutation relations. In order to study the full implications on the concept of locality, it is crucial to find the physical states of then maximal localization. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in posi- tions and in momenta. © 1996 American Institute of Physics. @S0022-2488~96!00305-3# I. INTRODUCTION The short distance structure of conventional geometry can be considered experimentally con- firmed up to the order of 1 TeV ~see, e.g., Ref. 1!. In string theory and quantum gravity certain corrections to the short distance structure and the uncertainty relations have been suggested to appear at smaller scales ~the latest at the Planck scale! ~see, e.g., Refs. 2–7 and, for a recent review, Ref. 8!. Here we continue a series of articles9–15 in which are studied the quantum theoretical conse- quences of small corrections to the canonical commutation relations @xi , pj#5i\~d i j1a i jklxkxl1b i jklpkpl1••• !, ~1! including the possibility that also @xi , xj#Þ0, @pi , pj#Þ0. A crucial feature of this ‘‘noncommu- tative geometric’’ ansatz, which was first studied in Ref. 11, is that for appropriate matrices a and b, Eq. ~1! implies the existence of finite lower bounds to the determination of positions and momenta. These bounds take the form of finite minimal uncertainties Dx0 and Dp0 , obeyed by all physical states. In fact, the approach covers the case of those corrections to the uncertainty relations which we mentioned above ~see Ref. 12!. A framework with a finite minimal uncertainty Dx0 can as well be understood to describe effectively nonpointlike particles than as describing a fuzzy space. As discussed in Refs. 12–15, the approach, with appropriately adjusted scales, could have therefore more generally a potential for an effective description of nonpointlike particles, such as, e.g., nucleons or quasi-particles in solids. a!Electronic mail address: fehaye@wicc.weizmann.ac.il b!Electronic mail address: a.kempf@amtp.cam.ac.uk 0022-2488/96/37(5)/2121/17/$10.00 2121J. Math. Phys. 37 (5), May 1996 © 1996 American Institute of Physics Downloaded¬13¬Apr¬2011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions Analogously, on large scales a minimal uncertainty Dp0 may offer new possibilities to de- scribe situations where momentum cannot be precisely determined, in particular on curved space.13 Using the path integral formulation it has been shown in Ref. 13 that such noncommutative background geometries can ultraviolet and infrared regularize quantum field theories in arbitrary dimensions through minimal uncertainties Dx0 ,Dp0 . However, a complete analysis of the modi- fied short distance structure, and, in particular, the calculation of the states of maximal localiza- tion, has so far only been carried out for the special case of the commutation relations @x, p#5i\~11bp2!, in Ref. 15. The reason is that those cases are representation theoretically much easier to handle in which either a or b vanish, i.e., with minimal uncertainties in either position or in momenta only. We now solve the more general, though still one-dimensional, problem involv- ing both minimal uncertainties in positions and in momenta. We define the associative Heisenberg algebra A with corrections parametrized by small constants a,b>0, @x, p#5i\~11ax21bp2!, ~2! or, in a notation which will prove more convenient ~q>1!, @x, p#5i\F11~q221 !S x24L2 1 p 2 4K2D G , ~3! where the constants L ,K carry units of length and momentum and are related by 4KL5\~11q2!. ~4! While the first correction term contributes for large ^x2&5^x&21(Dx)2, which is the definition of the infrared, the second correction term contributes for large ^p2&5^p&21(Dp)2, i.e., in the ultra- violet. The corresponding uncertainty relation DxDp> \ 2 $11a„~Dx ! 21^x&2…1b„~Dp !21^p&2…% ~5! holds in all *-representations of the commutation relations and reveals these infrared and ultra- violet modifications as minimal uncertainties in positions and momenta:12 ~Dxmin! 25L2 q221 q2 F11~q221 !S ^x& 2 4L2 1 ^p&2 4K2 D G , ~6! ~Dpmin!25K2 q221 q2 F11~q221 !S ^x& 2 4L2 1 ^p&2 4K2 D G . ~7! In particular, for all physical states, i.e., for all uc&PD with D,H being any *-representation of the commutation relations of A in a Hilbert space H , there are finite absolutely smallest uncer- tainties ~all uc& normalized!: ~Dx uc&!5^cu~x2^x&! 2uc&1/2>LA12q22 ;uc&PD , ~8! ~Dp uc&!5^cu~p2^p&!2uc&1/2>KA12q22 ;uc&PD . ~9! We will here only deal with the kinematical consequences of possible corrections to the commu- tation relations. Arbitrary systems can be considered and studies on dynamical systems, including 2122 H. Hinrichsen and A. Kempf: Minimal x,p—Uncertainties and localization J. Math. Phys., Vol. 37, No. 5, May 1996 Downloaded¬13¬Apr¬2011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions the calculation of the spectra of Hamiltonians and integral kernels such as Green’s functions, have been carried out for example systems in Refs. 9 and 10. Comparison can also be made with the features of the discretized quantum mechanics studied, e.g., in Refs. 16–18. An interesting ca- nonical field theoretical approach with a similar motivation is focusing on generalizing the uncer- tainty relations among the coordinates.19 II. HILBERT SPACE REPRESENTATION A crucial consequence of Eqs. ~8! and ~9! is that there are no eigenvectors to x nor to p in any space of physical states, i.e., in any *-representation D of the generalized commutation relations. As is clear from the definition of uncertainties, e.g., (Dx) uc&2 5^cu~x2^cuxuc&!2uc&, eigenvectors to x or p could only have vanishing uncertainty in position or momentum. In particular, the com- mutation relations of A no longer find spectral representations of x nor of p. In the situation of a50 ~or b50!, i.e., with Dp050 ~or Dx050!, there is still the momentum ~or position! representation of A available, in which case the maximal localization states have been calculated in Ref. 15. Let us now perform the analogous studies for the general case with a,b.0, where position and momentum space representations are both ruled out. To this end we use a Hilbert space representation of A on a generalized Fock space. The position and momentum operators can be represented as x5L~a†1a !, p5iK~a†2a !, ~10! where the a and a† obey generalized commutation relations aa†2q2a†a51 ~11! and act on the domain D of physical states D:5$uc&5polynomial~a†!u0&% as au0&50, a†un&5A@n11#un11& , ~12! aun&5A@n#un21&, where [n] denotes the partial geometric sum or ‘‘q’’-number @n#5 q2n21 q221 , ~13! and where the un&:5([n]!)21/2(a†)nu0&, n51,...,`, are orthonormalized, ^n1un2&5dn1 ,n2, ~14! and D is analytic and dense in the Hilbert space H5l2. While x and p ordinarily are essentially self-adjoint, they are now merely symmetric, which is sufficient to insure that all expectation values are real. The deficiency indices of x and p are ~1,1!, implying the existence of one-parameter families of self-adjoint extensions. While ordinarily self-adjoint extensions, e.g., for a particle in a box, need to and can be fixed, there is now the subtle effect of the self-adjoint extensions not being on common domains, which prevents the diagonalization of x or p on physical states, as can also be understood through the uncertainty relations. For the full functional analytical details see Ref. 12, where these structures have first been found. We will come back to these functional analytical studies in Sec. VI where we will explicitly calculate the diagonalizations in H . They are of use for the calculation of inverses of x 2123H. Hinrichsen and A. Kempf: Minimal x,p—Uncertainties and localization J. Math. Phys., Vol. 37, No. 5, May 1996 Downloaded¬13¬Apr¬2011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions and p, which are not only needed to describe certain quantum mechanical potentials, but ulti- mately also to invert kinetic terms, e.g., of the form p22m2 to obtain propagators from the field theoretical path integral ~see Ref. 13!. III. MAXIMAL LOCALIZATION STATES The absence of eigenvectors of x or p in all *-representations D of the commutation relations physically implies the absence of absolute localizability in position or momentum, i.e., there are no physical states that would have Dx50 or Dp50. More precisely, the uncertainty relation, holding in all D , implies a ‘‘minimal uncertainty gap:’’ '” uc&PD:Dx uc&,Dx0 and '” uc&PD:Dp uc&,Dp0 . ~15! The state of maximal localization in position ucxml& with given position expectation x and vanishing momentum expectation is defined through ^cx mluxucx ml&5x , ^cx mlupucx ml&50, ~Dx ! uc x ml&5Dxmin . ~16! Explicitly the minimal uncertainty in position then reads ~Dx ! uc x ml& 2 5L2 q221 q2 S 11~q221 ! ^x& 2 4L2 D ~17! with the corresponding ~now not infinite! uncertainty in momentum: ~Dp ! uc x ml& 2 5K2 ~q211 !2 q2~q221 ! S 11~q221 ! ^x& 2 4L2 D . ~18! We focus on maximal localization in x; the case of maximal localization in p is fully analogous. As shown in Ref. 15, a state of maximal localization is determined by the equation „~x2^x&!1ia~p2^p&!…ucxml&50, ~19! where a5Dx/Dp . Inserting Eqs. ~17! and ~18! we obtain a5 L~q221 ! K~q211 ! , ~20! so that the condition reads S q211L ~x2^x&!1i q 221 K pD ucxml&50. ~21! A. Maximal localization states in the Fock basis In order to explicitly calculate those states that realize the now maximally possible localiza- tion we expand the ucxml& in the Fock basis, ucx ml&:5 1 N ~x ! (n50 ` q23n/2cn~x !un&, ~22! where the cn(x) are real coefficients and 2124 H. Hinrichsen and A. Kempf: Minimal x,p—Uncertainties and localization J. Math. Phys., Vol. 37, No. 5, May 1996 Downloaded¬13¬Apr¬2011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions N ~x !:5 ( n50 ` q23ncn 2~x ! ~23! is a normalization factor ~the inserted factors q23n/2 will be convenient later!. The condition for maximal localization Eq. ~21! reads in the Fock representation: F ~q211 !S a†1a2 xL D2~q221 !~a†2a !G ucxml&50. ~24! Inserting the ansatz equation ~22! we are led to the recursion relation q1q21 2L xcn~x !5 Aq21@n11#cn11~x !1Aq@n#cn21~x !. ~25! Together with c21~x !50 and c0~x !51, ~26! the coefficients cn(x) are therefore determined as polynomials of degree n in x . B. Relation to continuous q-Hermite polynomials The coefficients cn(x) are related to the so-called continuous q-Hermite polynomials. An excellent review on these and other q-orthogonal polynomials is Ref. 20. We use the notation of shifted q-factorials20 ~a;q2!n :5 ) k50 n21 ~12aq2k!, ~27! which obey the identity ~a;q2!n5~2a !nqn~n21 !~a21;q22!n . ~28! Furthermore, we define for later convenience j~x !:5 arcsinh~vx !ln q , x~ j !5 q j2q2 j 2v , ~29! where v:5 1 4L ~q1q 21!Aq221. ~30! The continuous q-Hermite polynomials Hn(zuq2) are defined through H21~zuq2!50, H0~zuq2!51, ~31! and the recurrence relation ~see Ref. 20! 2zHn~zuq2!5Hn11~zuq2!1~12q2n!Hn21~zuq2!. ~32! It is not difficult to check that this recursion relation can be brought into the form of the recursion relation equation ~25! for the coefficients cn(x), by expressing them in terms of the Hn(zuq2) as 2125H. Hinrichsen and A. Kempf: Minimal x,p—Uncertainties and localization J. Math. Phys., Vol. 37, No. 5, May 1996 Downloaded¬13¬Apr¬2011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions cn~x !5A qn@n#!~q221 !ni2nHn~ ivxuq2!. ~33! As shown in Ref. 20 the continuous q-Hermite polynomials Hn(zuq2) can be written as Hn~zuq2!5 ( k50 n S nk D q2ei~n22k !u, z5cos u , ~34! with the q-binomial coefficients S nk D q25 ~q2;q2!n ~q2;q2!k~q2;q2!n2k . ~35! Inserting Eq. ~34! into Eq. ~33! and replacing [n]! by @n#!5 ~2 !n~q2;q2!n ~q221 !n 5 qn 2 ~q22;q22!n ~q2q21!n ~36! yields cn~x !5 1 Aqn2~q22;q22!n i2n( k50 n S nk D q2ei~n22k !u, ivx5cos u . ~37! Because of ivx5 12(q j(x)2q2 j(x)) , we may also write eiu5iq j(x) and therefore obtain the follow- ing exact expression for the coefficients cn(x): cn~x !5 1 Aqn2~q22;q22!n ( k50 n S nk D q2~2 !kq ~n22k ! j~x !. ~38! We derive further useful properties of the cn(x). Classical limit: For q!1 the recursion relation equation ~25! reduces to x L cn~x !5 An11cn11~x !1Ancn21~x !. ~39! By substituting x5L&z and Hn(z)5An!2ncn(x) we obtain H0(z)51 and 2zHn~z !5Hn11~z !12nHn21~z !, ~40! which is the defining recursion relation for classical Hermite polynomials Hn(z). Thus the clas- sical limit of the polynomials cn(x) is given by lim q!1 cn~x !5 1 An!2n HnS xL& D . ~41! Representation by the formula of Rodriguez: As a short notation we write x( j) as x j . Then, introducing the q-difference operator Df ~x j!5 f ~x j11!2 f ~x j21! x j112x j21 , ~42! 2126 H. Hinrichsen and A. Kempf: Minimal x,p—Uncertainties and localization J. Math. Phys., Vol. 37, No. 5, May 1996 Downloaded¬13¬Apr¬2011¬to¬202.115.51.3.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright;¬see¬http://jmp.aip.org/about/rights_and_permissions the polynomials cn(x j) can be expressed as cn~x j!5 ~2 !n kn q j 2 Dnq2 j 2 , ~43! where kn5Aq2n 2 @n#!S q1q212L D n . ~44! Equation ~43! generalizes the formula of Rodriguez, Hn(x)5(2)nex 2(dn/dxn)e2x2, for classical Hermite polynomials. A proof for Eq. ~43! is outlined in Appendix A. q-difference equation: The generalized formula of Rodriguez equation ~43! implies that cn~x j11!2cn~x j21!5Aq~12q22n!~q j1q2 j!cn21~x j!, ~45! which generalizes the differentiation rule (d/dx)Hn(x)5nHn21(x) for classical Hermite polyno- mials. In order to prove this equation, we rewrite its lhs using Eqs. ~42! and ~43!: cn~x j11!2cn~x j21!5~x j112x j21!Dcn~x j!5~x j112x j21!D ~2 !n kn q j 2 Dnq2 j 2 . ~46! Carrying out the first differentiation on the rhs of this formula @c.f. Eq. ~A1!#, one obtains a linear combination of the polynomials cn(x j) and cn11(x j), which in turn can be expressed through the recurrence relation equation ~25! in terms of cn21(x j). It can also be shown by induction that Eq. ~45! implies the following q-difference equation for the polynomials cn(x); q jcn~x j21!1q2 jcn~x j11!5q2n~q j1q2 j!cn~x j!, ~47! which corresponds to the differential equation 2xHn8(x)2Hn9(x)52nHn(x) for classical Hermite polynomials. Orthogonality: The polynomials cn(x) obey the orthogonality relation (j52` ` ~x2 j1k112x2 j1k21!q2~2 j1k! 2 cm~x2 j1k!cn~x2 j1k!5Nkqndm ,n , ~48! where Nk5 (j52` ` ~x2 j1k112x2 j1k21!q2~2 j1k! 2 . ~49! The parameter 0