Minimal areas from q-deformed oscillator algebras
aa r X i v : . [ h e p - t h ] J un Minimal areas
Minimal areas from q-deformed oscillator algebras
Andreas Fring • , Laure Gouba ◦ and Bijan Bagchi ∗ • Centre for Mathematical Science, City University London,Northampton Square, London EC1V 0HB, UK ◦ National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa ∗ Department of Applied Mathematics, University of Calcutta,92 Acharya Prafulla Chandra Road, Kolkata 700 009, IndiaE-mail: [email protected], [email protected],bbagchi123@rediffmail.com
Abstract:
We demonstrate that dynamical noncommutative space-time will give riseto deformed oscillator algebras. In turn, starting from some q-deformations of thesealgebras in a two dimensional space for which the entire deformed Fock space can beconstructed explicitly, we derive the commutation relations for the dynamical variables innoncommutative space-time. We compute minimal areas resulting from these relations,i.e. finitely extended regions for which it is impossible to resolve any substructure inform of measurable knowledge. The size of the regions we find is determined by thenoncommutative constant and the deformation parameter q. Any object in this type ofspace-time structure has to be of membrane type or in certain limits of string type.
1. Introduction
The idea to extend the quantization procedure from canonical variables to space-time itself[1] traces back over sixty years. In recent years this general possibility has become moreand more appealing, especially in the context of quantum field theories as such type ofspace-time structures will introduce natural cut-offs and theories on them are thereforerenormalized by construction [2, 3]. In addition, almost all possible theories of quantumgravity require non-Minkowskian space-time in one form or another [4, 5, 6, 7, 8].One of the interesting consequences of these type of space-time structures is that inmany cases they lead to modifications of Heisenberg’s uncertainty relations, which in turnresult in the emergence of minimal lengths. This means in such spaces one has almostinevitably definite fundamental distances below which no substructure can be resolved[9, 10, 11, 12, 13, 14, 15, 16, 17]. Recently some of us proposed [18] a consistent dynamicalnoncommutative space-time structure in a two dimensional space which leads to a funda-mental length in one direction, implying that objects in these spaces are of string type.Here we provide a different type of dynamical noncommutative space-time implying a fun-damental length in each of the two directions, thus giving rise to minimal areas for which inimal areas any substructures is beyond measurable knowledge. In our construction procedure we willnot only postulate the deformed Heisenberg canonical commutation relations and checktheir consistency, but we will also derive them from some more extensively studied andmore fundamental structure, namely q-deformed oscillator algebras for which the entireFock space can be constructed explicitly [12, 13, 14].In section 2 we commence with various consistent deformations of Heisenberg’s canon-ical commutation relations and investigate the consequences on the commutation relationsof the associated oscillator algebra. We find that the latter are almost inevitably deformed.In section 3 we take this fact into account and reverse the setting by starting instead froma well suited q-deformed oscillator algebra and derive from it Heisenberg’s uncertainty re-lations for the dynamical variables. In section 4 we briefly recall the standard argumentleading to minimal length and compute the minimal area for a selected algebra. Ourconclusions and an outlook to further open problems are stated in section 5.
2. Creation and annihilation operators from noncommutative space-time
Noncommutative flat space-time in two dimensions manifests itself in the following modi-fication of Heisenberg’s canonical commutation relations for the dynamical variables[ x , y ] = iθ, [ x , p x ] = i ~ , [ y , p y ] = i ~ , [ p x , p y ] = 0 , [ x , p y ] = 0 , [ y , p x ] = 0 . (2.1)Restricting the noncommutative constant to be real, i.e. θ ∈ R , ensures that x and y are Hermitian operators. We now wish to find a representation for creation and annihi-lation operators in terms of the dynamical variables x , y , p x , p y satisfying the standardcommutation relations for a Fock space representation[ a i , a † j ] = δ ij , [ a i , a j ] = 0 , [ a † i , a † j ] = 0 for i, j = 1 , . (2.2)In order to reduce the number of unknown coefficients in a possible Ansatz for the a i , a † i wemay take the properties of the dynamical variables under a PT -transformation as a guidingprinciple. These type of considerations have proved to be very fruitful, allowing even aconsistent formulation of non-Hermitian systems with real eigenvalues, see e.g. [19, 20, 21]for a review or [22, 23] for recent special issues. For this purpose we note that the relations(2.1) are P x T -symmetric and P y T -symmetric in the sense that they remain invariant undera simultaneous reflection in the x -direction together with a time reversal and under asimultaneous reflection in the y -direction together with a time reversal, respectively, P x : x
7→ − x , y y , p x
7→ − p x , p y p y , P y : x x , y
7→ − y , p x p x , p y
7→ − p y , T : x x , y y , p x
7→ − p x , p y
7→ − p y , i
7→ − i, P x T : x
7→ − x , y y , p x p x , p y
7→ − p y , i
7→ − i, P y T : x x , y
7→ − y , p x
7→ − p x , p y p y , i
7→ − i. (2.3)– 2 – inimal areas We demand now to have a definite transformation property for the a i , a † i , that is we wouldlike them to be either even or odd under a P x,y T -transformation, i.e. a i a i , a † i a † i or a i
7→ − a i , a † i
7→ − a † i , such that we can use this property to reduce the total number ofconstants. Assuming that the dependence on the x , y , p x , p y is still linear, the generaloperators of the form a := α x + iα y + iα p x + α p y , a † := α x − iα y − iα p x + α p y ,a := α x + iα y + iα p x + α p y , a † := α x − iα y − iα p x + α p y , (2.4)with unknown constants α , . . . , α ∈ R for the time being, are P x T -odd: a i
7→ − a i , a † i
7→ − a † i and P y T -even: a i a i , a † i a † i when using the realization (2.3). The reversescenario is simply achieved by α j iα j for j = 1 , . . . , α = α ~ ∆ , α = θα + ~ α ~ ∆ , α = − α ~ ∆ , α = − θα + ~ α ~ ∆ , (2.5)where we abbreviated ∆ := α α − α α = 0 . This means we have still four almost entirelyfree parameters left. Inverting the relations (2.4) while keeping the constraints (2.5), wecan express the coordinates and the momenta in terms of the creation and annihilationoperators x = ( θα + ~ α ) ( a + a † ) + ( θα + ~ α ) ( a + a † ) , y = iα ( a − a † ) − iα ( a − a † ) ,p x = − iα ( a − a † ) + iα ( a − a † ) , p y = − ~ α ( a + a † ) − ~ α ( a + a † ) . (2.6)It is easily verified that these operators obey (2.1) when using (2.2). Let us now carry out a similar analysis for the situation when the underlying space-time isdynamical, i.e. the constant θ becomes position and possibly also momentum dependent.A set of consistent commutation relations for such a scenario was introduced in [18][ x, y ] = iθ (1 + τ y ) , [ x, p x ] = i ~ (1 + τ y ) , [ y, p y ] = i ~ (1 + τ y ) , [ p x , p y ] = 0 , [ x, p y ] = 2 iτ y ( θp y + ~ x ) , [ y, p x ] = 0 . (2.7)Defining the analogues to the creation and annihilation operators and keeping the depen-dence on the dynamical variables similar as in (2.4)ˆ a := α x + iα y + iα p x + α p y , ˆ a † := α x − iα y − iα p x + α p y , ˆ a := α x + iα y + iα p x + α p y , ˆ a † := α x − iα y − iα p x + α p y , (2.8) For the specific choice α = α = − λ ~ √ K , α = − α = − √ K , α = − α = λ ~ √ K , α = α = 1 √ K , we recover the representation found in [24] when comparing with equations (57) and (58) therein andidentifying the quantities λ , λ and K , K which are defined in equation (56) and (59), respectively. – 3 – inimal areas we can compute the resulting commutation relations. Keeping the constraints (2.5) andsetting in addition α = 0 we find that the standard commutation relations are deformed[ˆ a i , ˆ a † i ] = 1 + τ α (cid:16) ˆ a ˆ a † + ˆ a † ˆ a − ˆ a ˆ a − ˆ a † ˆ a † (cid:17) for i = 1 , a , ˆ a ] = [ˆ a , ˆ a † ] = [ˆ a † , ˆ a ] = [ˆ a † , ˆ a † ] = τ α (cid:16) ˆ a ˆ a + ˆ a ˆ a † − ˆ a † ˆ a − ˆ a † ˆ a † (cid:17) . (2.10)The asymmetry between i = 1 and i = 2 in (2.9) appears odd at first sight in the lightof (2.8), but it is a consequence of the non-symmetric nature of (2.7) and our choice α = 0. Clearly when the deformation parameter τ vanishes we obtain the usual Fockspace commutation relations (2.2). We propose now a new type of deformation for the flat noncommutative space-time (2.1)[˜ x, ˜ y ] = iθ + iτ (cid:0) ˜ x + ˜ y (cid:1) , [˜ x, ˜ p x ] = i ~ + i τ ~ θ (cid:0) ˜ x + ˜ y (cid:1) , [˜ x, ˜ p y ] = 0 , [˜ p x , ˜ p y ] = iτ (cid:2) ~ θ (˜ y ˜ p x − ˜ x ˜ p y ) − ˜ p x − ˜ p y (cid:3) , [˜ y, ˜ p y ] = i ~ + i τ ~ θ (cid:0) ˜ x + ˜ y (cid:1) , [˜ y, ˜ p x ] = 0 . (2.11)In the same manner as for (2.7) we may verify that these commutation relations are con-sistent in the sense that the Jacobi identities are satisfied. Using the standard argumentsto find a minimal length, we observe that the ˜ x, ˜ y -commutator implies a minimal length inthe ˜ x as well as in the ˜ y -direction, which means the underlying object, whose substructurewe can not determine, is of a membrane structure. Once again we define creation andannihilation type operators analogously to (2.4) keeping the dependence on the dynamicalvariables the same. When specifying the coefficients such that˜ a := q − τ θ (˜ x + i ˜ y ) , ˜ a † := q − τ θ (˜ x − i ˜ y ) , ˜ a := q − τ θ (cid:2) ˜ x − i ˜ y + θ ~ (˜ p y + i ˜ p x ) (cid:3) , ˜ a † := q − τ θ (cid:2) ˜ x + i ˜ y + θ ~ (˜ p y − i ˜ p x ) (cid:3) , (2.12)we find the commutation relations˜ a i ˜ a † j − (cid:18) τ − τ (cid:19) δ ij ˜ a † j ˜ a i = δ ij , [˜ a † i , ˜ a † j ] = 0 , [˜ a i , ˜ a j ] = 0 , for i, j = 1 , . (2.13)As expected (2.2) is recovered for τ →
0. These relations are very reminiscent of theq-deformed oscillator algebra studied in this context for instance in [9, 10, 11, 12, 13, 14,15, 16].This example and the one in the previous subsection indicate that dynamical space-time relations will naturally lead to deformed Fock spaces. As we have seen some of themhave a very convenient and well studied structure, as (2.13), whereas others are ratherawkward such as (2.9) and (2.10). Let us therefore now reverse the scenario and deformfirst the Fock space relations in a “nice” way and subsequently compute the correspondingcommutation relations for the dynamical variables.– 4 – inimal areas
3. Noncommutative space-time from q-deformed creation and annihila-tion operators
Resembling the relations (2.13) we q -deform the relations in (2.2) by defining a new set ofcreation and annihilation operators A , A † , A , A † satisfying A i A † j − q δ ij A † j A i = δ ij , [ A † i , A † j ] = 0 , [ A i , A j ] = 0 , for i, j = 1 , . (3.1)There exist various other possibilities to deform the relations (2.2) which still lead toconstructable Fock spaces, such as for instance using different q s in the first relation of(3.1), i.e. q δ ij → q δ ij i or replacing the δ ij on the right hand side of the first relation by q g ( A † i A i ) with g ( x ) being an arbitrary function as in [11, 16]. Guided by the limit q → PT -transformation, we expand the new set of deformed canonical variables X, Y, P x , P y linearly in terms of the A , A † , A , A † as X = κ ( A † + A ) + κ ( A † + A ) , P x = iκ ( A † − A ) + iκ ( A † − A ) ,Y = iκ ( A † − A ) + iκ ( A † − A ) , P y = κ ( A † + A ) + κ ( A † + A ) . (3.2)The constants κ , . . . , κ ∈ R are unknown for the time being. Inverting the relations (3.2)we may express the deformed creation and annihilation operators in terms of the deformedcanonical variables A = κ λ X + i κ µ Y − i κ µ P x − κ λ P y , A † = κ λ X − i κ µ Y + i κ µ P x − κ λ P y ,A = − κ λ X − i κ µ Y + i κ µ P x + κ λ P y , A † = − κ λ X + i κ µ Y − i κ µ P x + κ λ P y , (3.3)where we abbreviated λ := 2( κ κ − κ κ ) = 0 and µ := 2( κ κ − κ κ ) = 0. Using therepresentation (3.2) together with (3.1) we compute[ X, Y ] = 2 i ( κ κ + κ κ ) + 2 i ( q − κ κ A † A + κ κ A † A ) , (3.4)[ X, P x ] = 2 i ( κ κ + κ κ ) + 2 i ( q − κ κ A † A + κ κ A † A ) , (3.5)[ Y, P y ] = − i ( κ κ + κ κ ) + 2 i (1 − q )( κ κ A † A + κ κ A † A ) , (3.6)[ P x , P y ] = − i ( κ κ + κ κ ) + 2 i (1 − q )( κ κ A † A + κ κ A † A ) , (3.7)[ X, P y ] = 0 , (3.8)[ Y, P x ] = 0 . (3.9)Next we employ the relations (3.3) and evaluate A † A = κ λ X + κ µ Y + κ µ P x + κ λ P y − κ κ λ XP y − κ κ µ Y P x (3.10)+ i κ κ λµ [ X, Y ] + i κ κ λµ [ Y, P y ] − i κ κ λµ [ X, P x ] − i κ κ λµ [ P x , P y ] ,A † A = κ λ X + κ µ Y + κ µ P x + κ λ P y − κ κ λ XP y − κ κ µ Y P x (3.11)+ i κ κ λµ [ X, Y ] + i κ κ λµ [ Y, P y ] − i κ κ λµ [ X, P x ] − i κ κ λµ [ P x , P y ] . – 5 – inimal areas Substituting (3.10) and (3.11) into the right hand sides of (3.4)-(3.7) we obtain four equa-tions for the four unknown commutators [
X, Y ], [
X, P x ], [ Y, P y ] and [ P x , P y ]. Solving theseequations, the resulting dynamical noncommutative relations are[ X, Y ] = iθ + i q − q − q + q − (cid:20) κ κ κ + κ κ κ ( κ κ − κ κ ) X + κ κ κ + κ κ κ ( κ κ − κ κ ) Y (3.12)+ κ κ ( κ κ + κ κ )( κ κ − κ κ ) P x + κ κ ( κ κ + κ κ )( κ κ − κ κ ) P y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) XP y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) Y P x (cid:21) , [ X, P x ] = i ~ + i q − q − q + q − (cid:20) κ κ κ + κ κ κ ( κ κ − κ κ ) X + κ κ ( κ κ + κ κ )( κ κ − κ κ ) Y (3.13)+ κ κ κ + κ κ κ ( κ κ − κ κ ) P x + κ κ ( κ κ + κ κ )( κ κ − κ κ ) P y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) XP y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) Y P x (cid:21) , [ Y, P y ] = i ~ − i q − q − q + q − (cid:20) κ κ ( κ κ + κ κ )( κ κ − κ κ ) X + κ κ κ + κ κ κ ( κ κ − κ κ ) Y (3.14)+ κ κ ( κ κ + κ κ )( κ κ − κ κ ) P x + κ κ κ + κ κ κ ( κ κ − κ κ ) P y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) XP y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) Y P x (cid:21) , [ P x , P y ] = − i q − q − q + q − (cid:20) κ κ ( κ κ + κ κ )( κ κ − κ κ ) X + κ κ ( κ κ + κ κ )( κ κ − κ κ ) Y (3.15)+ κ κ κ + κ κ κ ( κ κ − κ κ ) P x + κ κ κ + κ κ κ ( κ κ − κ κ ) P y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) XP y − κ κ ( κ κ + κ κ )( κ κ − κ κ ) Y P x (cid:21) . For the constant terms of these commutators we have implemented here the constraints κ κ + κ κ = θ (cid:0) q (cid:1) , (3.16) κ κ + κ κ = ~ (cid:0) q (cid:1) , (3.17) κ κ + κ κ = − ~ (cid:0) q (cid:1) , (3.18) κ κ + κ κ = 0 , (3.19)in order to ensure that the limit q → inimal areas Keeping all the constants generic in the algebra (3.12)-(3.15) will make the handling verycumbersome. However, using the fact that we still have four κ s free at our disposal allowsus to extract some special limiting cases in order to obtain some more tractable algebras. Considering (3.2) the first natural limit is to reduce the number of free parameters to four,e.g. κ , . . . , κ , and introduce some dependence for the coefficients in the Y -direction onthose in the X -direction. Considering the representation (3.3) we impose κ = κ , κ = − κ , κ = − κ and κ = κ , (3.20)such that without activating the constraints (3.16)-(3.19) the eight unknown constants arealready limited to four. The four constraints (3.16)-(3.19) are not independent for thesechoices as (3.17) and (3.18) become identical. The remaining three constraints read κ − κ = θ (cid:0) q (cid:1) , κ κ + κ κ = ~ (cid:0) q (cid:1) and κ = κ , (3.21)which means we have still one constant at our disposal. The algebra (3.12)-(3.15), (3.8)and (3.9) simplifies to[ X, Y ] = iθ + i q − q − q + q − (cid:20) κ κ − κ κ κ κ + κ κ ( X + Y ) − κ κ κ κ + κ κ ( XP y − Y P x ) (cid:21) , (3.22)[ X, P x ] = ih + i q − q − q + q − (cid:20) κ κ κ κ + κ κ ( X + Y ) + κ κ κ κ + κ κ ( P x + P y ) (cid:21) , (3.23)[ Y, P y ] = ih + i q − q − q + q − (cid:20) κ κ κ κ + κ κ ( X + Y ) + κ κ κ κ + κ κ ( P x + P y ) (cid:21) , (3.24)[ P x , P y ] = − i q − q − q + q − (cid:20) κ κ − κ κ κ κ + κ κ ( P x + P y ) − κ κ κ κ + κ κ ( XP y − Y P x ) (cid:21) , (3.25)[ X, P y ] = 0 , (3.26)[ Y, P x ] = 0 . (3.27)The conditions λ = 0, µ = 0 now coincide and have translated into κ κ + κ κ = 0. Ourchoice of constants has achieved that the terms XP y and Y P x have combined into theangular momentum operator L z . As one of the κ s is still not fixed we can simplify the commutation relations (3.22)-(3.27)further by setting κ = 0, such that all three unknown left are fixed by the remaining threerelations κ = θ (cid:0) q (cid:1) , κ κ = ~ (cid:0) q (cid:1) and κ = κ . (3.28)– 7 – inimal areas We may now implement the constraints (3.28) in the algebra (3.22)-(3.27) and eliminateall constants κ i being left with a purely q -deformed algebra[ X, Y ] = iθ + i q − q − q + q − (cid:0) X + Y (cid:1) , (3.29)[ X, P x ] = i ~ + i q − q − q + q − ~ θ (cid:0) X + Y (cid:1) , (3.30)[ Y, P y ] = i ~ + i q − q − q + q − ~ θ (cid:0) X + Y (cid:1) , (3.31)[ P x , P y ] = i q − − qq − + q (cid:20) P x + P y + 2 ~ θ ( XP y − Y P x ) (cid:21) , (3.32)[ X, P y ] = 0 , (3.33)[ Y, P x ] = 0 . (3.34)These relations reduce to (2.11) for q = ± p (1 + τ ) / (1 − τ ). Notice further that the q -deformation and the θ -deformation originally introduced in the space-space commutationrelations have become intrinsically linked through the constraints. We can no longer takethe limit θ → q →
0. However, the limit q → X and Y direction in a simultaneous measurement as we will explainin more detail below. As it stands, the relation (3.29) will lead to the same minimal lengthin either direction. This is by no means unavoidable and can be overcome by taking anotherlimit of the algebra (3.12)-(3.15), (3.8) and (3.9). Setting for instance κ = κ = 0 withoutany additional constraints besides (3.16)-(3.19), which in this case read κ κ = θ (cid:0) q (cid:1) , κ κ = ~ (cid:0) q (cid:1) , κ κ = − ~ (cid:0) q (cid:1) , κ κ = − κ κ . (3.35)the algebra simplifies considerably[ X, Y ] = iθ + i q − q − q + q − (cid:18) κ κ X + κ κ Y (cid:19) , (3.36)[ X, P x ] = i ~ + i q − q − q + q − (cid:18) κ κ X + κ κ κ Y (cid:19) , (3.37)[ Y, P y ] = i ~ − i q − q − q + q − (cid:18) κ κ κ X + κ κ Y (cid:19) , (3.38)[ P x , P y ] = − i q − q − q + q − (cid:20) ( κ κ + κ κ ) (cid:18) κ κ κ X + κ κ κ Y (cid:19) (3.39)+ κ κ P x + κ κ P y − κ κ κ κ Y P x − κ κ κ κ XP y (cid:21) , [ X, P y ] = 0 , (3.40)[ Y, P x ] = 0 . (3.41)We notice that in (3.36) we have now different coefficients in front of the X and Y -termsand may achieve unequal minimal length in either direction, although they are not entirelyindependent being related by the first relation in (3.35).– 8 – inimal areas Taking now a less trivial limit, we may obtain string like relations from (3.36)-(3.41)similar to those proposed in [18]. Parameterizing q = e τκ with τ ∈ R + and taking thelimit κ → κ = ~ /θκ , κ = ~ /θκ , κ = (1 + q ) / (4 κ ) and derive the simple “string type”relations[ X, Y ] = iθ (cid:0) τ Y (cid:1) , [ X, P x ] = i ~ (cid:0) τ Y (cid:1) , [ X, P y ] = 0 , [ P x , P y ] = iτ ~ θ Y , [ Y, P y ] = i ~ (cid:0) τ Y (cid:1) , [ Y, P x ] = 0 . (3.42)Arguing in the same way as in [18], we obtain now from the first relation in (3.42) a minimallength in the Y -direction in a simultaneous X, Y -measurement as the commutator [
X, Y ]is identical. The remaining commutators are, however, different.There are of course plenty of other possible limits compatible with the constraints(3.16)-(3.19), which we do not present here.
4. Minimal areas and minimal lengths
As mentioned, one of the interesting physical consequences of noncommutative space-time,especially when it is dynamical, is the emergence of minimal lengths in simultaneous mea-surements of two observables. The standard noncommutative space-time relations (2.1)give rise to additional uncertainties similar to the usual Heisenberg uncertainty relations,meaning for instance that the two position operators x and y can never be known withcomplete precision at the same time , where θ plays the role of ~ when compared with theconventional relations. When the underlying algebra becomes a dynamical noncommuta-tive space-time structure the consequences are more severe and one finds that the positionoperators X or Y can never be known, that is even when giving up the entire knowledgeabout the canonical conjugate partner Y or X , respectively. Thus X or Y are said tobe bound by some absolute minimal length ∆ X or ∆ Y , which is the highest possibleprecision to which these quantities can be resolved.Minimal lengths have been known and studied for some time [9, 10, 11, 12, 13, 14,15, 16] in simultaneous x, p -measurements as a consequence of a deformation of the x, p -commutator. In [18] it was demonstrated explicitly that they also result in simultaneous x, y -measurements as a consequence of the dynamical noncommutativity of space-time.Whereas the algebra investigated in [18] only gave rise to a minimal length in one direction,i.e. “string like” objects, we demonstrate here that the algebras provided in section 3 willlead to minimal lengths in two direction, i.e. minimal areas. Objects in these type ofspaces are “membrane like”, meaning that there exists a finitely extended region aboutwhose substructure it is impossible to obtain any measurable knowledge.Following the standard arguments we will now compute these quantities by startingwith the well known relation ∆ A ∆ B ≥ |h [ A, B ] i| , (4.1)which holds for any two observables A and B , which are Hermitian with respect to thestandard inner product. In order to determine the range of validity for this inequality– 9 – inimal areas we simply have to minimize f (∆ A, ∆ B ) := ∆ A ∆ B − |h [ A, B ] i| as a function of ∆ B tofind the absolute minimal length ∆ A . This means we need to solve the two equations ∂ ∆ B f (∆ A, ∆ B ) = 0 and f (∆ A, ∆ B ) = 0 for ∆ A =: ∆ A min and subsequently computethe smallest value for ∆ A min in order to obtain the absolute minimal length ∆ A . Incase we obtain minimal length for both of these observables we define the minimal areaand its smallest possible value of four times the product, that is ∆( AB ) min and ∆( AB ) ,respectively.For definiteness we choose now θ ∈ R + and carry out the analysis for the algebra(3.36)-(3.41) starting with a simultaneous X, Y -measurement. When q > X, Y ] are positive due to the first constraint in(3.35). The absolute value for |h [ X, Y ] i| is therefore simply Im h [ X, Y ] i . When q < | A − B | ≥ A − B for A, B > h X i is given by ∆ X = (cid:10) X (cid:11) − h X i and similarly for X ↔ Y , we compute∆ X min = q | q − | ( κ h X i + κ h Y i ) + θ ( q − κ κ qκ , (4.2)∆ Y min = q | q − | ( κ h X i + κ h Y i ) + θ ( q − κ κ qκ , (4.3)such that the absolute minimal lengths result to∆ X = κ q p | q − | and ∆ Y = κ q p | q − | , (4.4)hen h X i = h Y i = 0. Together with the first constraint in (3.35) the absolute minimal areain the X, Y -plane results to ∆( XY ) = θ (cid:12)(cid:12) q − q − (cid:12)(cid:12) . (4.5)This means the size of the minimal area is independent of the free parameters κ and κ .We can also make ∆ Y a function of ∆ X and compute for given ∆ X the correspondingminimal length ∆ Y or vice versa. Note that it is impossible to achieve any of the minimallengths to vanish without the other becoming infinitely large. We illustrate this in figure1, where we plot ∆ Y (∆ X ) = ± θ (cid:12)(cid:12) q − q − (cid:12)(cid:12) / (4∆ X ) for a specific value of θ and variousvalues of q . The two minimal areas indicated in the figure have the same size.For a simultaneous X, P x -measurement we compute similarly the minimal momentumin the X -direction(∆ P x ) min = q ( q − ( h Y i + h Y i ) κ κ + ~ | q − | κ κ κ + h X i ( q − κ κ ( q + 1) κ κ , (4.6)such that the corresponding absolute value turns out to be(∆ P x ) = 2 κ p | q − | q + 1 . (4.7)– 10 – inimal areas There is no minimal length for X in this case as we can tune ∆ X to be as small as we wishby enlarging ∆ P x . -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-14-12-10-8-6-4-202468101214 Y X q = 5 q = 2 Figure 1: Minimal areas in the XY-plane.
Similarly we compute for a simultaneous
Y, P y -measurement the minimal momentumin the Y -direction(∆ P y ) min = q ( q − ( h X i + h X i ) κ κ + ~ | − q | κ κ κ + h Y i ( q − κ κ ( q + 1) κ κ , (4.8)with corresponding absolute value(∆ P y ) = 2 κ p | q − | q + 1 . (4.9)By the same reasoning as in the previous case there is also no minimal length for Y in thiscase as ∆ Y can be taken to be as small as desiredh by enlarging ∆ P y .The analysis for a simultaneous P x , P y -measurement is less straightforward due to theappearance of the angular momentum term. we first note that |h [ P x , P y ] i| ≥ (cid:12)(cid:12)(cid:12)(cid:12) q − q + 1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) | κ κ + κ κ | (cid:18) κ κ κ (cid:10) X (cid:11) − (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ (cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Y (cid:11)(cid:19) (4.10)+ κ κ (cid:10) P x (cid:11) + κ κ (cid:10) P y (cid:11) − κ κ κ κ |h Y P x i| − κ κ κ κ |h XP y i| (cid:21) , where for definiteness we assumed that κ < κ . Using next the estimate |h AB i| ≤ ∆ A ∆ B + |h A i h B i| we compute∆ P x ∆ P y ≥ (cid:12)(cid:12)(cid:12)(cid:12) q − q + 1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) κ κ ∆ P x + κ κ ∆ P y − (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ κ (cid:12)(cid:12)(cid:12)(cid:12) ∆ Y ∆ P x − κ κ κ κ ∆ X ∆ P y + λ (cid:21) , (4.11)– 11 – inimal areas with λ = κ κ h P x i + κ κ h P y i + | κ κ + κ κ | (cid:18) κ κ κ h X i − (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ (cid:12)(cid:12)(cid:12)(cid:12) h Y i (cid:19) (4.12) − (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ κ (cid:12)(cid:12)(cid:12)(cid:12) |h Y i h P x i| − κ κ κ κ |h X i h P y i| . When varying the inequality (4.11) in the same manner as the expressions above we find(∆ P x ) min = − (cid:12)(cid:12) q − (cid:12)(cid:12) q κ κ κ κ ∆ X − (cid:0) q − (cid:1) q (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ κ (cid:12)(cid:12)(cid:12)(cid:12) κ κ ∆ Y (4.13) ± (cid:12)(cid:12) q − q − (cid:12)(cid:12) vuut κ κ ∆ X κ κ + κ κ ∆ Y κ κ + 2 (cid:12)(cid:12)(cid:12) κ κ κ κ (cid:12)(cid:12)(cid:12) κ ∆ X ∆ Yκ | q − | ( q + 1) − + 4 q λκ κ ( q − . and(∆ P y ) min = − (cid:0) q − (cid:1) q κ κ κ κ ∆ X − (cid:12)(cid:12) − q (cid:12)(cid:12) q (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ κ (cid:12)(cid:12)(cid:12)(cid:12) κ κ κ ∆ Y (4.14) ± (cid:12)(cid:12) q − q − (cid:12)(cid:12) vuut κ κ ∆ X κ κ + κ κ ∆ Y κ κ + 2 (cid:12)(cid:12)(cid:12) κ κ κ κ (cid:12)(cid:12)(cid:12) κ ∆ X ∆ Yκ ( q − | − q | − + 4 q λκ κ ( q − . We can minimize this expression further with a subsequent
X, Y -measurement. This is,however, a matter of interpretation if one would like to view measurements as a pairwisesuccession or whether this should be considered as a simultaneous measurement of fourquantities. A further option would be to exploit the explicit occurrence of the L z -operatorand take this complication here as a hint that the angular momentum variables are possiblya more natural set of variables. We leave this problem for future investigations. Similarexpressions are obtained for the choice κ > κ .
5. Conclusions
We have demonstrated that dynamical noncommutative space-time relations will inevitablylead to deformed oscillator algebras. Taking some well studied oscillator algebras with theuseful property that the entire Fock spaces associated to them is explicitly constructable asa starting point, we derived some very general commutation relations (3.12)-(3.15) for thedynamical variables. Since these relations are rather cumbersome, we investigated somespecific limits leading to simplified and more tractable variants, whose properties can bediscussed more transparently. All of these special limits led to minimal lengths in the twodimensional space and mostly to minimal areas which we have calculated explicitly (4.5).There are some obvious further problems following from our considerations. First ofall it would be very interesting to explore the consequences of taking different types ofdeformations as starting points and derive the resulting dynamical commutation relations.Secondly it would be interesting to consider explicit models on these type space-time struc-tures and thirdly but not last a generalization to three dimensional space would be highlyinteresting. The latter will almost inevitably lead to minimal volumes.– 12 – inimal areas
Acknowledgments:
A.F. would like to thank the UGC Special Assistance Programme inthe Applied Mathematics Department of the University of Calcutta and S.N. Bose NationalCentre for Basic Sciences for providing infrastructure and financial support. Thanks forextremely kind hospitality go to many members of these institutions, but especially toBijan Bagchi and Partha Guha for being tireless in this effort. L.G. is supported under thegrant of the National Research Foundation of South Africa.
References [1] H. S. Snyder, Quantized space-time, Phys. Rev. , 38–41 (1947).[2] M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. ,977–1029 (2001).[3] R. J. Szabo, Quantum Field Theory on Noncommutative Spaces, Phys. Rept. , 207–299(2003).[4] X. Calmet, M. Graesser, and S. D. H. Hsu, Minimum Length from Quantum Mechanics andClassical General Relativity, Phys. Rev. Lett. (21), 211101 (Nov 2004).[5] D. Gross and P. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B303 , 407(1988).[6] D. Amati, M. Ciafaloni, and G. Veneziano, Can Space-Time Be Probed Below the StringSize?, Phys. Lett.
B216 , 41 (1989).[7] D. Amati, M. Ciafaloni, and G. Veneziano, Higher order gravitational deflection and softBremsstrahlung in Planckian energy superstring collisions, Nucl. Phys.
B347 , 550–580(1990).[8] A. Ashtekar, Mathematical Problems of Non-perturbative Quantum General Relativity, LesHouches summer school 1992 on Gravitation and Quantization (1993).[9] A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry, J.Math. Phys. , 4483–4496 (1994).[10] A. Kempf, G. Mangano, and R. B. Mann, Hilbert space representation of the minimal lengthuncertainty relation, Phys. Rev. D52 , 1108–1118 (1995).[11] G. Brodimas, A. Jannussis, and R. Mignani, Bose realization of a noncanonical Heisenbergalgebra, J. Phys.
A25 , L329–L334 (1992).[12] L. C. Biedenham, The quantum group group SU (2) q and a q-analogue of the bosonoperators, J. Phys. A22 , L873–L878 (1989).[13] A. J. Macfarlane, On q-analogues of the quantum harmonic oscillator and the quantumgroup SU (2) q , J. Phys. A22 , 4581–4588 (1989).[14] C.-P. Su and H.-C. Fu, The q-deformed boson realisation of the quantum group SU ( n ) q andits representations, J. Phys. A22 , L983–L986 (1989).[15] C. Quesne and V. M. Tkachuk, Generalized deformed commutation relations with nonzerominimal uncertainties in position and/or momentum and applications to quantum mechanics,SIGMA , 016 (2007). – 13 – inimal areas [16] B. Bagchi and A. Fring, Minimal length in Quantum Mechanics and non-HermitianHamiltonian systems, Phys. Lett. A373 , 4307–4310 (2009).[17] S. Hossenfelder, Self-consistency in theories with a minimal length, Class. Quant. Grav. ,1815–1821 (2006).[18] A. Fring, L. Gouba, and F. G. Scholtz, Strings from dynamical noncommutative space-time,arXiv:1003.3025 .[19] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. , 947–1018(2007).[20] A. Mostafazadeh, Pseudo-Hermitian Quantum Mechanics, arXiv:0810.5643, to appear Int. J.Geom. Meth. Mod. Phys .[21] P. E. G. Assis, Non-Hermitian Hamiltonians in Field Theory, PhD thesis, City UniversityLondon (2010).[22] A. Fring, H. Jones, and M. Znojil (guest editors), Special issue dedicated to the physics ofnon-Hermitian operators (PHHQP VI) (City University London, UK, July 2007), J. Phys. A24 (June, 2008).[23] S. Jain and Z. Ahmed (guest editors), Non Hermitian Hamitonians in Quantum Physics -Part I and II (PHHQP VIII) (Bhabha Atomic Research Centre, India, January 2009),Pramana Journal of Physics (August, September, 2009).[24] F. G. Scholtz, L. Gouba, A. Hafver, and C. M. Rohwer, Formulation, Interpretation andApplication of non- Commutative Quantum Mechanics, J. Phys. A42 , 175303 (2009)., 175303 (2009).