ADDITIONAL DEGREES OF FREEDOM ASSOCIATED WITH POSITION MEASUREMENTS IN NON-COMMUTATIVE QUANTUM MECHANICS C. M. ROHWER

Thesis presented in partial fulfilment of the requirements for the degree of MASTER OF SCIENCE at the University of Stellenbosch. Supervisor : Professor F.G. Scholtz September 2010

DECLARATION I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.

Signature

Date

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Abstract Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a priori what is meant by a position measurement on a non-commutative space. It was shown recently in a paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanical position measurements consistently on the non-commutative plane. To do this, it is necessary to introduce weak (non-projective) measurements, formulated in terms of Positive Operator-Valued Measures (POVMs). In this thesis we shall demonstrate, however, that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the aforementioned formalism entails a description that is non-local in that it requires knowledge of all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary (in addition to position) to specify quantum states completely. The remainder of the thesis, based in part on a recent publication (“Noncommutative quantum mechanics – a perspective on structure and spatial extent”, C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic field found by Susskind. It will be argued through various applications that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, where the natural interpretation is that of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions are convenient, in that they make explicit the additional structure which is encoded more subtly in the non-local description. Lastly we shall argue that the additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory.

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Opsomming As gevolg van die minimum lengteskaal wat deur nie-kommuterende ko¨ordinate ge¨ınduseer word is dit nie a priori duidelik wat met ’n posisiemeting op ’n nie-kommutatiewe ruimte bedoel word nie. Dit is onlangs in ’n artikel deur Scholtz et al. getoon dat dit wel op ’n nie-kommutatiewe vlak moontlik is om die begrip van kwantummeganiese posisiemetings te herwin. Vir hierdie doel benodig ons die konsep van swak (nie-projektiewe) metings wat in terme van ’n positief operator-waardige maat geformuleer word. In hierdie tesis sal ons egter toon dat ’n meting van slegs die posisie nie volledige inligting oor die kwantumtoestand van ’n deeltjie in ’n niekommutatiewe ruimte lewer nie. Ons formalisme behels ’n nie-lokale beskrywing waarbinne kennis oor alle ordes van posisieafgeleides in die sogenaamde sterproduk bevat word. Die sterproduk is ’n welbekende konstruksie waardeur operatorvermenigvuldiging op funksievermenigvuldiging afgebeeld kan word. Ons sal toon dat verskeie ekwivalente lokale beskrywings bestaan wat volg uit die invoer van bykomende vryheidsgrade. Dit beteken dat nie-kommutatiewe posisiemetings op ’n natuurlike wyse die nodigheid van bykomende strukture uitwys wat noodsaaklik is om die kwantumtoestand van ’n sisteem volledig te beskryf. Die res van die tesis, wat gedeeltelik op ’n onlangse publikasie (“Noncommutative quantum mechanics – a perspective on structure and spatial extent”, C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) gebaseer is, behels ’n ondersoek na die fisiese interpretasie van hierdie bykomende strukture. Ons sal toon dat vir ’n spesifieke lokale formulering die beeld van objekte met struktuur op ’n natuurlike wyse in die ooreenstemmende klassieke teorie na vore kom. Hierdie beskrywing is inderdaad ekwivalent aan die van Susskind wat twee gelaaide deeltjies, gekoppel deur ’n harmoniese interaksie, in ’n sterk magneetveld behels. Met behulp van verskeie toepassings sal ons toon dat hierdie interpretasie op ’n natuurlike wyse na die kwantummeganiese konteks vertaal waar sekere dwangvoorwaardes na vore kom. ’n Tweede lokale beskrywing in terme van objekte wat by ’n sekere punt met ’n vaste hoekmomentum gelokaliseer is sal ook ondersoek word. Binne hierdie konteks sal ons weer deur die begrip van addisionele struktuur gekonfronteer word. Beide lokale beskrywings is gerieflik omdat hulle hierdie bykomende strukture eksplisiet maak, terwyl dit in die nie-lokale beskrywing deur die sterproduk versteek word. Laastens sal ons toon dat die bykomende vryheidsgrade in lokale beskrywings ook as ykvryheidsgrade van ’n ykinvariante formulering van die teorie beskou kan word.

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Acknowledgements I would like to express my sincerest thanks to my supervisor, Professor F.G. Scholtz. Due to the interpretational slant of this thesis, there were frequently periods where the path forward was unclear. For his accommodating support in identifying sensible questions and for his open door when the corresponding answers were evasive, I am most grateful. The Wilhelm Frank Bursary Trust, administrated by the Department of Bursaries and Loans at Stellenbosch University, provided financial support for my studies, not only during my M.Sc. but also during my B.Sc. and B.Sc. Hons. degrees. This aid was instrumental in allowing me to focus on academic priorities and I am thankful for the privilege of having received it. The many conversations with my fellow students A. Hafver and H.J.R. van Zyl were of immeasurable value and helped to shed light on several matters. I appreciate not only the academic aspects of these exchanges, but also the friendship and sound-board for voicing frustrations. A great word of thanks is due to Mr. J.N. Kriel for helping me with technical and calculational issues and for engaging in countless interpretational discussions. His many hours of patient and involved assistance are valued sincerely. For their hospitality at the S.N. Bose National Centre for Basic Sciences, Kolkata, and at the Centre for High Energy Physics of the Indian Institute of Science, Bangalore, I am most grateful to Professors B. Chakraborty and S. Vaidya, respectively. The visit to India during 2010 provided a valued forum for exchanging ideas and identifying interesting problems for future work. Lastly, and perhaps most importantly, I would like to thank my parents for their unconditional support and love. Their patience and understanding mean a great deal to me.

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CONTENTS Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Opsomming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

Background and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK . .

1

1.1

A unitary representation of the Heisenberg algebra . . . . . . . . . . . . . . . . .

1

1.2

The postulates of standard quantum mechanics . . . . . . . . . . . . . . . . . . .

2

1.3

Weak measurement: the language of Positive Operator Valued Measures . . . . .

5

2. THE FORMALISM OF NON-COMMUTATIVE QUANTUM MECHANICS . . . . .

9

2.1

A unitary representation of the non-commutative Heisenberg algebra . . . . . . .

9

2.2

Position measurement in non-commutative quantum mechanics: the need for a revised probabilistic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3. THE RIGHT SECTOR AND BASES FOR LOCAL POSITION MEASUREMENTS .

19

3.1

Arbitrariness of the right sector in non-local position measurements

. . . . . . .

19

3.2

Decomposition of the identity on Hq . . . . . . . . . . . . . . . . . . . . . . . . .

20

3.3

POVMs for local position measurements . . . . . . . . . . . . . . . . . . . . . . .

23

4. THE BASIS |z, v) ≡ T (z) |0i hv| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.1

Decomposition of the identity on Hq and the associated POVM . . . . . . . . . .

25

4.2

An analysis of the corresponding classical theory . . . . . . . . . . . . . . . . . .

27

4.3

Constraints and differential operators on (z, v|ψ) . . . . . . . . . . . . . . . . . .

30

4.4

Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

4.5

Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.6

Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5. THE BASIS |z, n) ≡ T (z) |0i hn| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

5.1

A basis with a discrete right sector label — interpretation and probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

5.2

Relating the states |z, v) and |z, n) . . . . . . . . . . . . . . . . . . . . . . . . . .

46

5.3

Average energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

5.4

Some probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

50

5.4.1

Pure position measurements in terms of the non-local POVM (2.32) . . .

50

5.4.2

Position measurements in terms of the local POVM (5.13) – probing the right sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Some transition probabilities between states . . . . . . . . . . . . . . . . .

53

6. THE RIGHT SECTOR VIEWED AS GAUGE DEGREES OF FREEDOM . . . . . .

56

5.4.3

6.1

A gauge-invariant formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

6.2

A local transformation of the right sector seen as a gauge transformation

. . . .

58

6.3

Adding dynamics for the gauge field – some cautious speculations . . . . . . . . .

59

7. DISCUSSION AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

A. Inclusion of a third co-ordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

B. Proof of equation (2.35) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

C. The path integral action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

D. Momentum eigenstates (4.36) as a complete basis for Hq . . . . . . . . . . . . . . . . .

69

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

vii

LIST OF FIGURES 4.1

A schematic showing the two-charge (harmonically coupled) composite, whose orbital motion affects its shape deformation and vice versa. . . . . . . . . . . . . . . .

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35

Background and Motivations As strange as the idea of introducing non-commutative spatial co-ordinates into quantum mechanical theories may seem, it is certainly not as novel as one may expect. In fact, suggestions that space-time co-ordinates may be non-commutative appeared already in the early days of quantum mechanics. For instance, in his article [1] of 1947, Snyder pointed out that it is problematic to describe matter and local interactions relativistically in continuous 4-dimensional space-time due to the appearance of divergences in field theories in this context. It is shown there that there exists Lorentz-invariant space-time in which there is a natural unit of length, the introduction of which partially remedies aforementioned divergences. It is also demonstrated that the notion of a smallest unit of length can only be implemented upon having dropped assumptions of commutative space-time: commuting co-ordinates would have continuous spectra which would contradict the idea of spatial quantisation. More recently, the notion of non-commutative spacetime was investigated also from the perspective of gravitational instabilities. In [2], Dopplicher et al. argued that attempts at spatial localisation with precision smaller than the Planck length, r `p =

G~ ' 1.6 × 10−33 cm, c3

(1)

result in the collapse of gravitational theories in that they would require energy concentrations large enough to induce black hole formation. A natural solution to this problem would be to impose a minimum bound on localisability. On an intuitive level, one may understand this to be a consequence of the fact that a minimal length scale implies a regularisation of high momenta (through the Fourier transformation), which in turn restricts the attainable energy concentrations. Since non-commuting operators induce uncertainty relations, a natural way to impose such a bound on localisability is to introduce co-ordinates that do not commute. By finding a Hilbert space representation of a non-commuting algebra, these authors then introduced the concept of optimal localisation and put forward first steps toward field theories in this context. At this point, already, one may ask whether the notion of a point particle makes any sense in a space with finite, non-zero minimal bounds on spatial localisability. Though the answer to this question is far from obvious, it is clear that a local description of a point particle, i.e., one where we allocate a specific position to a point particle which has no physical extent, is nonsensical if we cannot specify co-ordinates to arbitrary accuracy. One fundamental motivation behind the study of frameworks such as string theory, is the need for a consistent field theoretical framework for extended objects where the notion of point-like local interactions may be replaced ix

by non-local interactions. In this particular context, it was shown by Susskind et al. that a free particle moving in the non-commutative plane can be thought of as two oppositely charged particles interacting through a harmonic potential and moving in a strong magnetic field [3] — an idea which makes the notion of physical extent and structure quite explicit. This article also alludes to the important role that non-commutative geometries play in the framework of string theory. Seiberg explains in [4] that the extended nature of strings leads to ambiguities in defining geometry and topology of space-time, and that field theories on non-commutative space in fact correspond to low-energy limits of string theories. Indeed, the study of field theories on noncommutative geometries – another setting in which non-local interactions occur quite naturally – has grown into a sizeable research field of its own; for an extensive review, see, for instance, [5]. Furthermore, the framework of non-commutative geometry provides a useful mathematical setting for the study of matrix models in string theory, as set out in [6]. In the context of the Landau problem, non-commutativity of guiding center co-ordinates in the lowest Landau level is well-known (the non-commutative parameter here scales inversely to the magnitude of the magnetic field); a detailed discussion can be found in [7]. Thermodynamic quantities such as the entropy of a non-commutative fermion gas have also been shown to exhibit non-extensive features due to the excluded volume effects induced at high densities by non-commutativity [8]. Non-commutative geometry appears in various other physical applications – a comprehensive summary may be found in [9].

Returning to the issues discussed earlier in this chapter, we see from many arguments there that the standard views of space-time merit further scrutiny and possibly even drastic revision. Evidently one candidate for addressing many of the problems encountered in this context is the introduction of non-commutative spatial co-ordinates. We have also seen that the issues of locality of measurements and the notion of extendedness go hand-in-hand with such modified space-time frameworks. Indeed, a consistent probability framework to describe position measurements in non-commutative quantum mechanics — a matter which is not trivial since non-commuting co-ordinates do not allow for simultaneous eigenstates — was formulated in [10]. This thesis departs with a detailed investigation of the non-locality1 of this description. We shall then proceed to introduce a manifestly local description for non-commutative quantum mechanical position measurements on a generic level, and subsequently focus on two specific choices of basis and their interpretations. As stated above, local position measurements of point 1 With non-locality we mean that this description requires knowledge not only of the position wave function, but also of all orders of spatial derivatives thereof. This non-locality is encoded in the so-called star product, and will be elaborated upon in the chapters to follow.

x

particles do not make sense in non-commutative space. Consequently it is not surprising that aforementioned local descriptions require the introduction of additional degrees of freedom and that constraints arise in some of these formulations. At this point it would be quite natural to ask whether non-commutative quantum mechanics might allow for an interpretation in terms of objects with additional structure and / or extent. Indeed, it has been shown (see Section 2 of [23]) that the conserved energy and total angular momentum derived from the non-commutative path integral action in [22] contain explicit correction terms to those for a point particle. Thus we see that, already on a classical level, there are hints at structured objects in a non-commutative theory. To provide further motivation for this standpoint, we shall show that the physical picture of Susskind [3] that was mentioned above appears explicitly in the non-commutative classical theory corresponding to one of our local descriptions. In this context we shall also demonstrate that the aforementioned correction terms to the conserved energy may also be formulated in terms of the additional degrees of freedom of this local description. Through application of this formulation to eigenstates of angular momentum, the free particle and the quantum harmonic oscillator we shall demonstrate that Susskind’s view is natural also in the context of non-commutative quantum mechanics. A further local formulation will be shown to allow a natural interpretation in terms of objects with an angular momentum about a point of localisation. Naturally such a point of view is incompatible with that of a point-particle whose internal degrees of freedom have not been specified. We shall conclude that the notion of additional structure is undeniably present in any such local description, and, most importantly, that complete information about non-commutative quantum mechanical states cannot be provided by a measurement of position only.

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CHAPTER 1 A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK In the following two chapters we shall discuss in detail the formalism that will be used for the remainder of this thesis. In order to illustrate the consequences of introducing non-commutative coordinates, we first review standard quantum mechanics, focusing on the significance of algebraic commutation relations and the statistical interpretations of measurement processes. Thereafter the non-commutative formalism as set out in [10] will be introduced and described in detail, with particular attention payed to the identification of measurable quantities and a suitable framework for position measurements. Note that all analyses will be restricted to two dimensions, i.e., our formalism applies to a non-commutative plane.2

1.1

A unitary representation of the Heisenberg algebra

The cornerstone of standard quantum mechanics is the set of canonical commutation relations. The relevant underlying structure is the abstract Heisenberg algebra, which reads [x, y] = 0, [x, px ] = [y, py ] = i~,

(1.1)

[px , py ] = [x, py ] = [y, px ] = 0. The generators of the algebra are linked to observable quantities through the construction of a unitary representation in terms of Hermitian operators that act on the quantum mechanical Hilbert space. The states of the system are represented by vectors in this quantum Hilbert space, which shall henceforth be denoted by Hq . These operators obviously obey the same commutation relations as those above, [ˆ x, yˆ] = 0, [ˆ x, pˆx ] = [ˆ y , pˆy ] = i~,

(1.2)

[ˆ px , pˆy ] = [ˆ x, pˆy ] = [ˆ y , pˆx ] = 0. 2

In Appendix A we discuss briefly the inclusion of a third co-ordinate and the associated problems pertaining to transformation properties and rotational invariance in higher dimensions.

1

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK

2

Two representations are common in the setting of standard quantum mechanics — the Schr¨odinger representation and Heisenberg’s matrix representation.3 For the former, for instance, we have that the position and momentum operators act on the Hilbert space of square-integrable wave functions as follows: x ˆψ(x, y) = xψ(x, y), ∂ pˆx ψ(x, y) = −i~ ψ(x, y), ∂x

(1.3)

and similarly for y.

The commutation relations (1.2) induce an uncertainty in the position and momentum observables, ∆ˆ x∆ˆ px ≥ where we define ∆Aˆ ≡

~ , 2

∆ˆ y ∆ˆ py ≥

~ , 2

(1.4)

q ˆ 2 for any observable A. ˆ 4 On a physical level, (1.4) simply hAˆ2 i − hAi

implies that, for a given direction, momentum and position cannot be measured simultaneously to arbitrary accuracy. In contrast to this, however, the two co-ordinates may be measured simultaneously since x and y commute in (1.1), as do the operators (1.2) representing them on the Hilbert space. It is this particular feature that will later be altered in a non-commutative setting.

Having reviewed the matter of representations of the abstract Heisenberg algebra, let us revisit the statistical interpretation associated with measurements in the standard quantum mechanical formalism.

1.2

The postulates of standard quantum mechanics

In standard quantum mechanics, measurements are considered to be projective. To illustrate precisely what is meant by this statement, we now recap the fundamental postulates of this probabilistic framework. We shall follow the discussion of [12], where the quantum mechanical formulation of von Neumann (see, for instance, [13]) is summarised.

Postulates of Standard Quantum Mechanics: I To every quantum mechanical observable we assign a corresponding Hermitian operator, 3 From the Stone-von Neumann theorem we know that all unitary representations of the algebra (1.1) are equivalent; see, for instance, [14]. 4 The proof hereof is simple, and relies on the Schwartz inequality; see, for instance, [11].

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK

3

A = A† . Due to the Hermiticity of A, we can construct a complete orthonormal basis (which, for simplicity, we assume here to be discrete) for Hq from the eigenvectors of A: A |φn i = λn |φn i , hφn |φm i = δn,m

⇒

Hq = spann {|φn i}.5

(1.5)

Naturally A has a spectral representation in terms of these eigenvectors: X

A =

λn |φn i hφn |

n

X

≡

λ n Pn .

(1.6)

n

The Hermiticity of A also guarantees a real spectrum, λn ∈ < ∀ n. II We call the operators Pn ≡ |φn i hφn | projectors. They sum to the identity on the quantum Hilbert space, X

P n = 1q .

(1.7)

n

Since the eigenvectors of A are orthogonal (see (1.5)), we have that Pn Pm = |φn i hφn |φm i hφm | = δn,m Pn .

(1.8)

Consequently any projector squares to itself, i.e., Pn2 = Pn . This implies that its eigenvalues must be 0 or 1. III A measurement of the observable A must necessarily yield one of the eigenvalues of A, say λα ∈ {λn | n = 0 : ∞}. If the system is originally in a normalised pure state |ψi, then the probability6 of measuring λα is given by pα = |hφα |ψi|2 = hψ| Pα |ψi = hψ| Pα2 |ψi = |Pα |ψi|2 .

(1.9)

These probabilities are non-negative and sum to unity,

pα ≥ 0,

X

pα = 1

⇒

0 ≤ pα ≤ 1,

(1.10)

α 5

At this point we do not stipulate the dimensionality of Hq . The inherent randomness in the measurement process becomes manifest in this postulate: we are not guaranteed any particular outcome. The only prediction we can make is the set of possible outcomes, and to each element thereof we may assign a probability. It is in this context that the notion of ensemble measurements is a natural interpretation. 6

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK

4

as is required for any probability. (These statements are easily verified using equations (1.6), (1.7) and (1.9)). The normalised state of the system after measurement is Pα |ψi |φi ≡ p . hψ| Pα |ψi

(1.11)

If another measurement is performed immediately on the system, it is clear from (1.6) and (1.8) that the outcome will again be λα with a probability of 1. It is in this sense that we consider measurements to be projective, since repeated measurements of a particular observable will yield the same result, i.e., the system is projected into a particular eigenstate of the observable in the measurement process. For the case where the system is initially in a mixed state described by the density operator ρ, the probability of measuring outcome λα is pα = trq (Pα ρPα ) = trq (Pα2 ρ) = trq (Pα ρ),

(1.12)

and the corresponding post-measurement state is described by the density operator ρα =

Pα ρPα Pα ρPα = . trq (Pα ρPα ) trq (Pα ρ)

(1.13)

(Here trq denotes the trace over the quantum Hilbert space, Hq ). IV The expectation value of A, in the sense of repeated measurements on an ensemble of identically prepared systems initially in state |ψi, is given by the probability-weighted sum of all possible outcomes, hAi =

X

pn λn .

(1.14)

n

The extension to a system initially in the mixed state ρ is simply hAi =

X

λn trq {Pn ρ} = trq {Aρ}.

(1.15)

n

Although the above postulates outline the usual approach to / interpretation of the statistical quantum mechanical framework, it is possible to relax some of these points. Indeed, the stipulation of projectivity in measurements is a very restrictive one, and we shall demonstrate in the following section that it is possible to build a consistent probabilistic framework where this

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK

5

requirement is relaxed.

1.3

Weak measurement: the language of Positive Operator Valued Measures

One of the most important underpinnings of quantum mechanics is the conservation of probability. This is guaranteed by insisting on Hermiticity of observables, which ensures unitarity in dynamic evolution of the system. Naturally such a description must be applied to closed quantum systems which are devoid of interactions with an environment that may violate conservation. Indeed, open quantum systems are typically described in terms of non-Hermitian operators that represent coupling to the environment.7 Generally such descriptions involve an alteration of the postulates set out above. As will be seen in later sections, it is necessary also in the framework of non-commutative quantum mechanical position measurements to modify the postulates of measurement slightly. For this reason we shall consider here a well-established extension to the statistical formalism above, that is of use not only in our framework but also in fields like quantum computing [16] and open quantum systems [17]. Returning to the matter at hand, we note that, to build a consistent probability framework, it is necessary to have a set of non-negative normalised probabilities as in (1.10). Looking at equation (1.9), we note that this is possible even if the operators Pn are not positive: it suffices to have positivity for Pn2 . We will show that this can be done even if one abandons the requirement of orthogonality (1.8) for the operators Pn which generate the post-measurement state (1.11). Suppose now that the normalised post-measurement state after a specific experiment, Dα |ψi , |φi = q hψ| Dα† Dα |ψi

(1.16)

is determined by a set of non-orthogonal operators {Dn }, Dn Dm 6= δn,m . We call these operators “detection operators”, and they are a generalisation of the orthogonal projectors Pn from (1.6). As an extension of the operators Pn2 , we further introduce a set of positive operators πn that sum to the identity on Hq , πn ≥ 0,

X

πn = 1q .

(1.17)

n

With this we have a so-called Positive Operator Valued Measure (POVM), where each πα is an element of the POVM. We note that one way to guarantee positivity of the POVM elements is through the identification πn = Dn† Dn . 7

For an example of such a description, see [15].

(1.18)

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK 1/2

The obvious choice of detection operator would be Dn = πn

6

(the square root of πn exists since

the operator is positive). However, the most general choice of detection operator satisfying (1.18) is Dn = Un πn1/2 ,

(1.19)

where Un is an arbitrary unitary transformation whose relevance will become clear shortly. Comparing the post-measurement states (1.11) and (1.16), we note that it would be natural to associate the α in Dα with a particular outcome of an observable (which need not be Hermitian). Let us proceed by introducing a modified set of postulates (based on the above POVMs) that allows the construction of a consistent, non-projective quantum mechanical probability interpretation for measurements of such quantities.

Modified Postulates of Quantum Mechanics: Non-Projective Measurements I We no longer require that the operators representing observables on Hq need be Hermitian. (This need not imply that Hermitian observables no longer exist, we simply do not demand Hermiticity of all observables). II Our point of departure is a decomposition of the identity on Hq in terms of positive operators (i.e. a POVM): πn ≥ 0 ∀ n,

X

π n = 1q .

(1.20)

n

The elements of the POVM may be decomposed further in terms of so-called detection operators, πn = Dn† Dn ,

(1.21)

where Dn† 6= Dn and Dn Dm 6= δn,m in general, but where X

Dn† Dn = 1q .

(1.22)

n

III A detection must necessarily yield an outcome corresponding to one of the elements of the POVM, say πα . If the system is originally in a normalised pure state |ψi, then the probability of this particular outcome is pα = |Dα |ψi|2 = hψ| Dα† Dα |ψi = hψ| πα |ψi . The previous postulate ensures that pα ≥ 0,

P

α pα

(1.23)

= 1, and consequently that 0 ≤ pα ≤ 1.

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK

7

For a system initially in a mixed state with density operator ρ, the probability for this particular outcome is pα = trq (Dα ρDα† ) = trq (Dα† Dα ρ) = trq (πα ρ).

(1.24)

Note that we do not prescribe the form of the operator corresponding to the observable quantity. Rather we consider the set of possible outcomes of a measurement, each being a label of a particular POVM. It need not be the case that these outcomes are necessarily eigenstates of a particular operator. IV The state of the system after measurement is Dα |ψi . |φi = q † hψ| Dα Dα |ψi

(1.25)

Recalling that the most general form of the detection operators is Dn = Un πn1/2 ,

(1.26)

we see that the state (1.25) can only be specified up to a unitary transformation, which induces a degree of arbitrariness after measurement. Consequently we cannot make any exact statements about the post-measurement state other than its norm. Furthermore, due to the non-orthogonality of the detection operators, a repeated measurement need not yield the same result (in contrast to (1.11)). It is in this sense that this framework describes non-projective measurements. For a mixed state ρ, the post-measurement state of the system is described by ρα =

Dα ρDα† trq (Dα ρDα† )

=

Dα ρDα† trq (Dα† Dα ρ)

.

(1.27)

V For any observable O, the expectation value is defined as hOi ≡ trq (Oρ).

(1.28)

This concludes our review of the standard quantum mechanical framework and the associated probabilistic formalism(s) for describing measurements. We now introduce the non-commutative

1. A REVIEW OF THE STANDARD QUANTUM MECHANICAL FRAMEWORK

8

framework by modifying the Heisenberg algebra (1.1), finding a suitable unitary representation on Hq , and discussing the implications of this formalism on position measurements.

CHAPTER 2 THE FORMALISM OF NON-COMMUTATIVE QUANTUM MECHANICS 2.1

A unitary representation of the non-commutative Heisenberg algebra

The framework of non-commutative quantum mechanics that we will use was put forward in [10], where a consistent probability interpretation for this formalism was outlined. Said article essentially comprises a consolidation and subsequent extension of the basic machinery used in [18] and [19]. Since this construction is vital to our analyses, we shall review these discussions thoroughly. The foundation of our construction is the introduction of a non-commutative configuration space.8 Co-ordinates on this space satisfy the commutation relation [ˆ x, yˆ] = iθ,

(2.1)

where θ is a real parameter (in units of length squared) that is assumed to be positive without loss of generality. By implication, the first line of the abstract Heisenberg algebra (1.1) is modified; its non-commutative analogue reads [x, y] = iθ, [x, px ] = [y, py ] = i~,

(2.2)

[px , py ] = [x, py ] = [y, px ] = 0. The task at hand is to find the quantum Hilbert space, Hq , and a unitary representation of the non-commutative algebra on this space. Returning to (2.1), we note that non-commutative co-ordinates cannot be scalars since these would commute. For this reason we denote the coordinates by hatted operators in (2.1). In order to find a basis for classical configuration space, it is convenient to define the following creation and annihilation operators:

b = b† =

1 √ (ˆ x + iˆ y ), 2θ 1 √ (ˆ x − iˆ y ). 2θ

8

(2.3)

We will investigate the case where the commutation relations of the momenta are unchanged, and only positional commutation relations are altered.

9

2. THE FORMALISM OF NON-COMMUTATIVE QUANTUM MECHANICS

10

It is easy to verify (using (2.1)) that these operators satisfy the Fock algebra [b, b† ] = 1.

(2.4)

This simply implies that the classical configuration space is isomorphic to boson Fock space, (b† )n ∼ Hc = F ≡ span |ni = √ |0i ; n = 0 : ∞ . n!

(2.5)

Consequently classical configuration space is a Hilbert space, which we shall denote by Hc . This is not an unusual feature, since standard commutative configuration space (i.e.,