Infinite-dimensional Categorical Quantum Mechanics

Transcription

1 Infinite-dimensional Categorical Quantum Mechanics A talk for CLAP Scotland Stefano Gogioso and Fabrizio Genovese Quantum Group, University of Oxford 5 Apr 2017 University of Strathclyde, Glasgow S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 1 / 24

2 Introduction - motivation for this work We want to do (diagrammatic) CQM in -dimensions, but... 1 Although there is a characterisation of orthonormal bases in terms of H -algebras. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 2 / 24

3 Introduction - motivation for this work We want to do (diagrammatic) CQM in -dimensions, but... Hilb has no unital -Frobenius algebras 1 NO group algebras NO Fourier sampling 1 Although there is a characterisation of orthonormal bases in terms of H -algebras. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 2 / 24

4 Introduction - motivation for this work We want to do (diagrammatic) CQM in -dimensions, but... Hilb has no unital -Frobenius algebras 1 NO group algebras NO Fourier sampling Hilb is not dagger compact NO traces, cups or caps NO operator-state duality 1 Although there is a characterisation of orthonormal bases in terms of H -algebras. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 2 / 24

5 Introduction - motivation for this work We want to do (diagrammatic) CQM in -dimensions, but... Hilb has no unital -Frobenius algebras 1 NO group algebras NO Fourier sampling Hilb is not dagger compact NO traces, cups or caps NO operator-state duality Hilb lacks other useful gadgets NO plane-waves or delta functions NO unbounded operators 1 Although there is a characterisation of orthonormal bases in terms of H -algebras. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 2 / 24

6 Introduction - motivation for this work We want to do (diagrammatic) CQM in -dimensions, but... Hilb has no unital -Frobenius algebras 1 NO group algebras NO Fourier sampling Hilb is not dagger compact NO traces, cups or caps NO operator-state duality Hilb lacks other useful gadgets NO plane-waves or delta functions NO unbounded operators Can we recover all of this (using non-standard analysis)? 1 Although there is a characterisation of orthonormal bases in terms of H -algebras. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 2 / 24

7 Introduction - motivation for this work We want to do (diagrammatic) CQM in -dimensions, but... Hilb has no unital -Frobenius algebras 1 NO group algebras NO Fourier sampling Hilb is not dagger compact NO traces, cups or caps NO operator-state duality Hilb lacks other useful gadgets NO plane-waves or delta functions NO unbounded operators Can we recover all of this (using non-standard analysis)? YES, WE CAN. 1 Although there is a characterisation of orthonormal bases in terms of H -algebras. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 2 / 24

8 Introduction - limit constructions, algebraically Non-standard analysis: an algebraic way to handle limit constructions 2. 2 Regardless of topological convergence. The sceptics out there might prefer to think directly in terms of the ultraproduct construction: we work in spaces of sequences, quotiented by a notion of asymptotic equality, or equality almost everywhere, determined by some non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 3 / 24

9 Introduction - limit constructions, algebraically Non-standard analysis: an algebraic way to handle limit constructions 2. (a) Natural numbers are unbounded, and hence: (i) infinite non-standard natural numbers exist (ii) any sequence has an non-standard extension to infinite natural indices 2 Regardless of topological convergence. The sceptics out there might prefer to think directly in terms of the ultraproduct construction: we work in spaces of sequences, quotiented by a notion of asymptotic equality, or equality almost everywhere, determined by some non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 3 / 24

10 Introduction - limit constructions, algebraically Non-standard analysis: an algebraic way to handle limit constructions 2. (a) Natural numbers are unbounded, and hence: (i) infinite non-standard natural numbers exist (ii) any sequence has an non-standard extension to infinite natural indices (b) Algebraic manipulation of series (without taking limits): (i) consider a sequence of partial sums a n := n j=1 b j (ii) extend it to obtain infinite sums ν j=1 b j, where ν is infinite natural 2 Regardless of topological convergence. The sceptics out there might prefer to think directly in terms of the ultraproduct construction: we work in spaces of sequences, quotiented by a notion of asymptotic equality, or equality almost everywhere, determined by some non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 3 / 24

11 Introduction - limit constructions, algebraically Non-standard analysis: an algebraic way to handle limit constructions 2. (a) Natural numbers are unbounded, and hence: (i) infinite non-standard natural numbers exist (ii) any sequence has an non-standard extension to infinite natural indices (b) Algebraic manipulation of series (without taking limits): (i) consider a sequence of partial sums a n := n j=1 b j (ii) extend it to obtain infinite sums ν j=1 b j, where ν is infinite natural (c) Some genuinely new finite vectors arise in non-standard Hilbert spaces: e.g. 1 ν { ν e n form an orthonormal basis e n, where ν is an infinite natural n=1 2 Regardless of topological convergence. The sceptics out there might prefer to think directly in terms of the ultraproduct construction: we work in spaces of sequences, quotiented by a notion of asymptotic equality, or equality almost everywhere, determined by some non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 3 / 24

12 Introduction - the Transfer Theorem The heavy lifting in non-standard analysis is done by the following result. Theorem (Transfer Theorem) A sentence ϕ holds in the standard model M of some theory with quantifiers ranging over standard elements, functions, relations and subsets if and only if the sentence ϕ holds in any/all non-standard models M of the theory with quantifiers ranging over internal non-standard elements, functions, relations and subsets. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 4 / 24

13 Introduction - the Transfer Theorem Example (Natural predecessors) Consider the sentence defining predecessors in the natural numbers: n N. [ n 0 [ m N. n = m + 1] ] By TT, the following sentence holds in the non-standard model N: n N. [ n 0 [ m N. n = m + 1] ] Hence all non-zero non-standard naturals have predecessors. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 5 / 24

14 Introduction - the Transfer Theorem Example (Well-ordering of naturals) Consider the sentence defining the well-order property for the natural numbers, i.e. saying that every non-empty subset of N has a minimum: A N. [A [ m A. a A.m a ]] By TT, the following sentence holds in the non-standard model N: [ A N. A [ m A. a A.m a ]] Hence all non-empty internal subsets A N have a minimum. (The requirement that A be internal is key here: e.g. the subset of all infinite non-standard naturals has no minimum, but it is also not internal.) S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 6 / 24

15 Introduction - the Transfer Theorem Example (Partial sums) Consider the sentence defining the sequence s : N R of partial sums for every sequence f : N R in the standard model R: f : N R. s : N R. [ s(0) = f (0) [ n N.s(n + 1) = s(n) + f (n + 1)] ] By TT, the following sentence holds in the non-standard model R: f : N R. s : N R. [ s(0) = f (0) [ n N.s(n + 1) = s(n) + f (n + 1)] ] Hence every internal sequence f : N R admits a corresponding internal sequence of partial sums s : N R, i.e. the notation m n=0 f (n) is legitimate for all m N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 7 / 24

16 The category Hilb - objects Objects are pairs H := ( H, P H ) specified by the following data: S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 8 / 24

17 The category Hilb - objects Objects are pairs H := ( H, P H ) specified by the following data: (i) a non-standard Hilbert space H (the underlying Hilbert space); S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 8 / 24

18 The category Hilb - objects Objects are pairs H := ( H, P H ) specified by the following data: (i) a non-standard Hilbert space H (the underlying Hilbert space); (ii) an internal non-standard linear map P H : H H such that: S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 8 / 24

19 The category Hilb - objects Objects are pairs H := ( H, P H ) specified by the following data: (i) a non-standard Hilbert space H (the underlying Hilbert space); (ii) an internal non-standard linear map P H : H H such that: P H is a self-adjoint idempotent (the truncating projector); S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 8 / 24

20 The category Hilb - objects Objects are pairs H := ( H, P H ) specified by the following data: (i) a non-standard Hilbert space H (the underlying Hilbert space); (ii) an internal non-standard linear map P H : H H such that: P H is a self-adjoint idempotent (the truncating projector); there are a non-standard natural D N and a family ( e d ) D d=1 of non-standard vectors in H (an orthonormal basis for H) such that P H = D e d e d d=1 S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 8 / 24

21 The category Hilb - objects Objects are pairs H := ( H, P H ) specified by the following data: (i) a non-standard Hilbert space H (the underlying Hilbert space); (ii) an internal non-standard linear map P H : H H such that: P H is a self-adjoint idempotent (the truncating projector); there are a non-standard natural D N and a family ( e d ) D d=1 of non-standard vectors in H (an orthonormal basis for H) such that P H = D e d e d By Transfer Theorem we have that D is unique, and we define the dimension of object H to be the non-standard natural dim H := D. d=1 S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 8 / 24

22 The category Hilb - morphisms Morphisms F : H K in Hilb are the those internal non-standard linear maps F : H K such that: P K F P H = F S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 9 / 24

23 The category Hilb - morphisms Morphisms F : H K in Hilb are the those internal non-standard linear maps F : H K such that: P K F P H = F In particular, the identity for an object H is the truncating projector: id H := P H S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 9 / 24

24 The category Hilb - morphisms Morphisms F : H K in Hilb are the those internal non-standard linear maps F : H K such that: P K F P H = F In particular, the identity for an object H is the truncating projector: id H := P H This makes Hilb a full subcategory of the Karoubi envelope for the category of non-standard Hilbert spaces and C-linear maps. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 9 / 24

25 The category Hilb - -symmetric monoidal structure Morphisms F : H K in Hilb can be expressed as matrices with non-standard dimensions, using orthonormal bases for H and K: F = dim K d =1 dim H d=1 e d F d d e d S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 10 / 24

26 The category Hilb - -symmetric monoidal structure Morphisms F : H K in Hilb can be expressed as matrices with non-standard dimensions, using orthonormal bases for H and K: F = dim K d =1 dim H d=1 e d F d d e d In particular, the identity on H can be expressed as follows: id H = dim H d=1 e d e d S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 10 / 24

27 The category Hilb - -symmetric monoidal structure Morphisms F : H K in Hilb can be expressed as matrices with non-standard dimensions, using orthonormal bases for H and K: F = dim K d =1 dim H d=1 e d F d d e d In particular, the identity on H can be expressed as follows: id H = dim H d=1 e d e d Equipped with Kronecker product, conjugate transpose, and the C-linear structure of matrices, Hilb is an enriched -symmetric monoidal category, with C as its field of scalars. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 10 / 24

28 The category Hilb - some classical structures If e d dim H d=1 is an orthonormal basis for H, the following comultiplication and counit define a unital special commutative -Frobenius algebra on H: := dim H d=1 e d e d e d := e d dim H d=1 S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 11 / 24

29 The category Hilb - some classical structures If e d dim H d=1 is an orthonormal basis for H, the following comultiplication and counit define a unital special commutative -Frobenius algebra on H: := dim H d=1 e d e d e d := e d dim H d=1 When e d dim d=1 H is the non-standard extension of a standard complete orthonormal basis e d d=1, the comultiplication is the non-standard extension of the standard isometry given by the H*-algebra associated with e d d=1. In that case, the counit is the genuinely non-standard object. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 11 / 24

30 The category Hilb - dagger compact structure (i) Consider an object H, and a decomposition P H = dim H d=1 e d e d of its truncating projector in terms of some orthonormal basis of H. 3 By Transfer Theorem, this definition is independent of the choice of basis. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 12 / 24

31 The category Hilb - dagger compact structure (i) Consider an object H, and a decomposition P H = dim H d=1 e d e d of its truncating projector in terms of some orthonormal basis of H. (ii) Let ξ d be the state in H corresponding to the effect e d in H. 3 By Transfer Theorem, this definition is independent of the choice of basis. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 12 / 24

32 The category Hilb - dagger compact structure (i) Consider an object H, and a decomposition P H = dim H d=1 e d e d of its truncating projector in terms of some orthonormal basis of H. (ii) Let ξ d be the state in H corresponding to the effect e d in H. (iii) The dual object is defined by H := ( H, P H ), where we let 3 : P H := dim H d=1 ξ d ξ d 3 By Transfer Theorem, this definition is independent of the choice of basis. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 12 / 24

33 The category Hilb - dagger compact structure (i) Consider an object H, and a decomposition P H = dim H d=1 e d e d of its truncating projector in terms of some orthonormal basis of H. (ii) Let ξ d be the state in H corresponding to the effect e d in H. (iii) The dual object is defined by H := ( H, P H ), where we let 3 : P H := dim H d=1 ξ d ξ d (iv) Cups and caps can then be defined as follows: dim H := ξ n e n := dim H n=1 n=1 e n ξ n 3 By Transfer Theorem, this definition is independent of the choice of basis. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 12 / 24

34 The category Hilb - dagger compact structure (i) Consider an object H, and a decomposition P H = dim H d=1 e d e d of its truncating projector in terms of some orthonormal basis of H. (ii) Let ξ d be the state in H corresponding to the effect e d in H. (iii) The dual object is defined by H := ( H, P H ), where we let 3 : P H := dim H d=1 ξ d ξ d (iv) Cups and caps can then be defined as follows: dim H := ξ n e n := dim H n=1 n=1 e n ξ n (v) The category-theoretic dimension for H is Tr P H = dim H. 3 By Transfer Theorem, this definition is independent of the choice of basis. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 12 / 24

35 Case study - wavefunctions with periodic boundary Wavefunctions in an n-dimensional box with periodic boundary conditions. (i) Underlying Hilbert space L 2 [ ( R/Z ) n ]. (ii) Complete orthonormal basis of momentum eigenstates: i2π k x χ k := x e (iii) Dimension D := (2ω + 1) n, where ω is some infinite natural. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 13 / 24

36 Case study - wavefunctions with periodic boundary Wavefunctions in an n-dimensional box with periodic boundary conditions. (i) Underlying Hilbert space L 2 [ ( R/Z ) n ]. (ii) Complete orthonormal basis of momentum eigenstates: i2π k x χ k := x e (iii) Dimension D := (2ω + 1) n, where ω is some infinite natural. Classical structure corresponding to the momentum observable: := +ω k 1= ω... +ω k n= ω χ k χ k χ k := +ω k 1= ω... +ω k n= ω χ k S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 13 / 24

37 Case study - wavefunctions with periodic boundary The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω +ω :=... χ k+h χ k χ h := χ 0 k 1,h 1= ω k n,h n= ω S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 14 / 24

38 Case study - wavefunctions with periodic boundary The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω +ω :=... χ k+h χ k χ h := χ 0 k 1,h 1= ω k n,h n= ω The addition used here is that of the abelian group Z n 2ω+1 : from the point of view of Z n, it is cyclic on { ω,..., +ω} n ; from the point of view of Z n, it cycles beyond infinity. In particular, it contains Z n as a proper subgroup. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 14 / 24

39 Case study - wavefunctions with periodic boundary The classical states for are those in the following form, where x takes the form x = 1 2ω+1 q for some q Z n 1 2ω+1 (i.e. we have x 2ω+1 Z n 2ω+1 ): δ x := +ω k 1 = ω... +ω k n= ω χ k (x) χ k S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 15 / 24

40 Case study - wavefunctions with periodic boundary The classical states for are those in the following form, where x takes the form x = 1 2ω+1 q for some q Z n 1 2ω+1 (i.e. we have x 2ω+1 Z n 2ω+1 ): δ x := +ω k 1 = ω... +ω k n= ω χ k (x) χ k The classical states for behave as Dirac deltas: δ x0 f f (x 0 ), for near-standard smooth f and near-standard x 0 We call them the position eigenstates, and the position observable. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 15 / 24

41 Interlude - approximating tori by periodic lattices The requirement that x 1 2ω+1 Z n 2ω+1 for position eigenstates δ x is a consequence of the fact that the functions χ k are multiplicative characters of Z n, but not necessarily of Z n 2ω+1. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 16 / 24

42 Interlude - approximating tori by periodic lattices The requirement that x 1 2ω+1 Z n 2ω+1 for position eigenstates δ x is a consequence of the fact that the functions χ k are multiplicative characters of Z n, but not necessarily of Z n 2ω+1. An undesirable extra phase e i2π(2ω+1)s x (for generic s j { 1, 0, +1}) appears when equation δ x = δ x δ x is expanded, and this phase cancels out in general if and only if x 1 2ω+1 Z n 2ω+1. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 16 / 24

43 Interlude - approximating tori by periodic lattices The requirement that x 1 2ω+1 Z n 2ω+1 for position eigenstates δ x is a consequence of the fact that the functions χ k are multiplicative characters of Z n, but not necessarily of Z n 2ω+1. An undesirable extra phase e i2π(2ω+1)s x (for generic s j { 1, 0, +1}) appears when equation δ x = δ x δ x is expanded, and this phase cancels out in general if and only if x 1 2ω+1 Z n 2ω+1. 1 From the non-standard point of view, 2ω+1 Z n 2ω+1 is a periodic lattice 1 of infinitesimal mesh 2ω+1 in the non-standard torus (R/Z) n. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 16 / 24

44 Interlude - approximating tori by periodic lattices The requirement that x 1 2ω+1 Z n 2ω+1 for position eigenstates δ x is a consequence of the fact that the functions χ k are multiplicative characters of Z n, but not necessarily of Z n 2ω+1. An undesirable extra phase e i2π(2ω+1)s x (for generic s j { 1, 0, +1}) appears when equation δ x = δ x δ x is expanded, and this phase cancels out in general if and only if x 1 2ω+1 Z n 2ω+1. 1 From the non-standard point of view, 2ω+1 Z n 2ω+1 is a periodic lattice 1 of infinitesimal mesh 2ω+1 in the non-standard torus (R/Z) n. 1 From the standard point of view, 2ω+1 Z n 2ω+1 approximates all elements of the standard torus (R/Z) n up to infinitesimal equivalence. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 16 / 24

45 Case study - wavefunctions with periodic boundary The position and momentum observables are strongly complementary, a manifestation of the Weyl Canonical Commutation Relations. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 17 / 24

46 Case study - wavefunctions with periodic boundary The position and momentum observables are strongly complementary, a manifestation of the Weyl Canonical Commutation Relations. Position observable defined by the group algebra for boosts B k. Momentum observable acts as the group algebra for translations T x : ( { } δ x x 1 ) ( Z n 2ω+1,, = 1 ) Z n 2ω + 1 2ω + 1 2ω+1, +, 0 S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 17 / 24

47 Case study - wavefunctions with periodic boundary The position and momentum observables are strongly complementary, a manifestation of the Weyl Canonical Commutation Relations. Position observable defined by the group algebra for boosts B k. Momentum observable acts as the group algebra for translations T x : ( { } δ x x 1 ) ( Z n 2ω+1,, = 1 ) Z n 2ω + 1 2ω + 1 2ω+1, +, 0 The Weyl Canonical Commutation Relations in graphical form: δ x χ k = δ x χ k χ k δ x T x B k χ k (x) B k T x S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 17 / 24

48 Case study - wavefunctions on lattices Wavefunctions on an n-dimensional lattice Z n. (i) Underlying Hilbert space L 2 [Z n ]. (ii) Complete orthonormal basis of position eigenstates: { 1 if k = h δ k := h 0 otherwise (iii) Dimension D := (2ω + 1) n, where ω is some infinite natural. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 18 / 24

49 Case study - wavefunctions on lattices Wavefunctions on an n-dimensional lattice Z n. (i) Underlying Hilbert space L 2 [Z n ]. (ii) Complete orthonormal basis of position eigenstates: { 1 if k = h δ k := h 0 otherwise (iii) Dimension D := (2ω + 1) n, where ω is some infinite natural. Classical structure corresponding to the position observable: := +ω k 1= ω... +ω k n= ω δ k δ k δ k := +ω k 1= ω... +ω k n= ω δ k S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 18 / 24

50 Case study - wavefunctions on lattices The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω +ω :=... δ k+h δ k δ h := δ 0 k 1,h 1= ω k n,h n= ω S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 19 / 24

51 Case study - wavefunctions on lattices The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω +ω :=... δ k+h δ k δ h := δ 0 k 1,h 1= ω k n,h n= ω Its classical states are those in the following form, for x 1 2ω+1 Z n 2ω+1 : χ x := +ω k 1 = ω... +ω k n= ω e i2π k x δ k We call them the momentum eigenstates (they are self-evidently plane-waves), and the momentum observable. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 19 / 24

52 Case study - wavefunctions on lattices The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω +ω :=... δ k+h δ k δ h := δ 0 k 1,h 1= ω k n,h n= ω Its classical states are those in the following form, for x 1 2ω+1 Z n 2ω+1 : χ x := +ω k 1 = ω... +ω k n= ω e i2π k x δ k We call them the momentum eigenstates (they are self-evidently plane-waves), and the momentum observable. Once again, position and momentum observables are strongly complementary. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 19 / 24

53 Interlude - approximating real space by lattices A common trick in non-standard analysis sees standard real space approximated by non-standard lattices of infinitesimal mesh. 4 For the sceptics out there: an odd non-standard natural κ N is an equivalence class κ = [(k i ) i N ] of sequences the elements of which are asymptotically odd, or odd almost everywhere, according to the chosen non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 20 / 24

54 Interlude - approximating real space by lattices A common trick in non-standard analysis sees standard real space approximated by non-standard lattices of infinitesimal mesh. (i) Fix two odd 4 infinite naturals ω uv, ω ir N. 4 For the sceptics out there: an odd non-standard natural κ N is an equivalence class κ = [(k i ) i N ] of sequences the elements of which are asymptotically odd, or odd almost everywhere, according to the chosen non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 20 / 24

55 Interlude - approximating real space by lattices A common trick in non-standard analysis sees standard real space approximated by non-standard lattices of infinitesimal mesh. (i) Fix two odd 4 infinite naturals ω uv, ω ir N. (ii) Write ω uv ω ir = 2ω + 1 for some (unique) infinite natural ω N. 4 For the sceptics out there: an odd non-standard natural κ N is an equivalence class κ = [(k i ) i N ] of sequences the elements of which are asymptotically odd, or odd almost everywhere, according to the chosen non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 20 / 24

56 Interlude - approximating real space by lattices A common trick in non-standard analysis sees standard real space approximated by non-standard lattices of infinitesimal mesh. (i) Fix two odd 4 infinite naturals ω uv, ω ir N. (ii) Write ω uv ω ir = 2ω + 1 for some (unique) infinite natural ω N. (iii) Consider the periodic lattice 1 ω uv Z n 2ω+1 of infinitesimal mesh in the non-standard torus ( R/ω ir Z) n. 4 For the sceptics out there: an odd non-standard natural κ N is an equivalence class κ = [(k i ) i N ] of sequences the elements of which are asymptotically odd, or odd almost everywhere, according to the chosen non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 20 / 24

57 Interlude - approximating real space by lattices A common trick in non-standard analysis sees standard real space approximated by non-standard lattices of infinitesimal mesh. (i) Fix two odd 4 infinite naturals ω uv, ω ir N. (ii) Write ω uv ω ir = 2ω + 1 for some (unique) infinite natural ω N. (iii) Consider the periodic lattice 1 ω uv Z n 2ω+1 of infinitesimal mesh in the non-standard torus ( R/ω ir Z) n. (iv) The standard reals R are recovered by restricting to the (aperiodic) 1 sub-lattice of finite elements ω uv Z n 2ω+1 ( R 0 /ω ir Z) n, and then quotienting by infinitesimal equivalence : R = ( 1 ω uv Z n 2ω+1 ( R 0 /ω ir Z) n)/ 4 For the sceptics out there: an odd non-standard natural κ N is an equivalence class κ = [(k i ) i N ] of sequences the elements of which are asymptotically odd, or odd almost everywhere, according to the chosen non-principal ultrafilter F on N. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 20 / 24

58 Case study - wavefunctions in real space Wavefunctions in n-dimensional real space R n. (i) Underlying Hilbert space L 2 [R n ]. (ii) Orthonormal set of non-standard momentum eigenstates: χ p := x 1 ωuv e i2π (p x), for all p 1 ω uv Z n 2ω+1 (iii) Dimension D := (2ω + 1) n, where 2ω + 1 = ω uv ω ir. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 21 / 24

59 Case study - wavefunctions in real space Wavefunctions in n-dimensional real space R n. (i) Underlying Hilbert space L 2 [R n ]. (ii) Orthonormal set of non-standard momentum eigenstates: χ p := x 1 ωuv e i2π (p x), for all p 1 ω uv Z n 2ω+1 (iii) Dimension D := (2ω + 1) n, where 2ω + 1 = ω uv ω ir. Classical structure corresponding to the momentum observable: := +ω ir... p 1= ω ir +ω ir p n= ω ir χ p χ p χ p := +ω ir... p 1= ω ir +ω ir p n= ω ir χ p S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 21 / 24

60 Case study - wavefunctions in real space The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω ir +ω ir :=... χ p+q χ p χ q := χ 0 p 1,q 1= ω ir p n,q n= ω ir S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 22 / 24

61 Case study - wavefunctions in real space The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω ir +ω ir :=... χ p+q χ p χ q := χ 0 p 1,q 1= ω ir p n,q n= ω ir Its classical states are those in the following form, for x 1 ω ir Z n 2ω+1 : δ x := +ω ir p 1 = ω ir... +ω ir p n= ω ir χ p (x) χ p Once again, the classical states for them the position eigenstates, and behave as Dirac deltas, so we call the position observable. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 22 / 24

62 Case study - wavefunctions in real space The following multiplication and unit define a unital quasi-special commutative -Frobenius algebra, with normalisation factor (2ω + 1) n : +ω ir +ω ir :=... χ p+q χ p χ q := χ 0 p 1,q 1= ω ir p n,q n= ω ir Its classical states are those in the following form, for x 1 ω ir Z n 2ω+1 : δ x := +ω ir p 1 = ω ir... +ω ir p n= ω ir χ p (x) χ p Once again, the classical states for behave as Dirac deltas, so we call them the position eigenstates, and the position observable. And once again the position and momentum observables are strongly complementary. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 22 / 24

63 More stuff out there, and a lot more to come The framework already covers a lot more material: quantum fields on infinite lattices (non-separable); quantum fields in real spaces (non-separable); quantum algorithm for the Hidden Subgroup Problem on Z n ; Mermin-type non-locality arguments for infinite-dimensional systems. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 23 / 24

64 More stuff out there, and a lot more to come The framework already covers a lot more material: quantum fields on infinite lattices (non-separable); quantum fields in real spaces (non-separable); quantum algorithm for the Hidden Subgroup Problem on Z n ; Mermin-type non-locality arguments for infinite-dimensional systems. And even more material is currently being worked out: position/momentum duality, quantum symmetries and dynamics; applications to other quantum protocols (e.g. RFI quantum teleport n); wavefunctions/fields over general locally compact abelian Lie groups; wavefunctions/fields over Minkowski space; connections with Feynman diagrams. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 23 / 24

65 Thank You! Thanks for Your Attention! Any Questions? S Gogioso, F Genovese. Infinite-dimensional CQM. arxiv: S Gogioso, F Genovese. Towards Quantum Field Theory in CQM 5. arxiv: v2 S Abramsky, C Heunen. H*-algebras and nonunital FAs. arxiv: A Robinson. Non-standard analysis. Princeton University Press, 1974 CQM := Categorical Quantum Mechanics FA := Frobenius algebra 5 This is a revised and extended version, and will be out by the end of the week. S Gogioso, F Genovese (Oxford) Infinite-dimensional CQM CLAP Scotland 24 / 24