Lecture 1 Preview: Quantum and Classical Mechanics

  …it was on a Sunday that the idea first occurred to me that ab− ba might correspond to a Poisson bracket.

 P.A.M. Dirac,

The following lectures describe some links between the group SL₂(ℝ), the Heisenberg group and hypercomplex numbers. The described relations appear in a natural way without any enforcement from our side. The discussion is illustrated by mathematical models of various physical systems.

In this chapter we will demonstrate that a Poisson bracket do not only corresponds to a commutator, in fact a Poisson bracket is the image of the commutator under a transformation which uses dual numbers.

1.1 Axioms of Mechanics

There is a recent revival of interest in foundations of quantum mechanics, which is essentially motivated by engineering challenges at the nano-scale. There are strong indications that we need to revise the development of the quantum theory from its early days.

In 1926, Dirac discussed the idea that quantum mechanics can be obtained from classical description through a change in the only rule, cf. [81]:

…there is one basic assumption of the classical theory which is false, and that if this assumption were removed and replaced by something more general, the whole of atomic theory would follow quite naturally. Until quite recently, however, one has had no idea of what this assumption could be.

In Dirac’s view, such a condition is provided by the Heisenberg commutation relation of coordinate and momentum variables [81]*(1):

Algebraically, this identity declares noncommutativity of q_r and p_r. Thus, Dirac stated [81] that classical mechanics is formulated through commutative quantities (“c-numbers” in his terms) while quantum mechanics requires noncommutative quantities (“q-numbers”). The rest of theory may be unchanged if it does not contradict to the above algebraic rules. This was explicitly re-affirmed at the first sentence of the subsequent paper [80]:

The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions.

The same point of view is expressed in his later works [83]*p. 6[82]*p. 26.

Dirac’s approach was largely approved, especially by researchers on the mathematical side of the board. Moreover, the vague version – “quantum is something noncommutative” – of the original statement was lightly reverted to “everything noncommutative is quantum”. For example, there is a fashion to label any noncommutative algebra as a “quantum space” [72].

Let us carefully review Dirac’s idea about noncommutativity as the principal source of quantum theory.

1.2 “Algebra” of Observables

Dropping the commutativity hypothesis on observables, Dirac made [81] the following (apparently flexible) assumption:

All one knows about q-numbers is that if z₁ and z₂ are two q-numbers, or one q-number and one c-number, there exist the numbers z₁ + z₂, z₁ z₂, z₂ z₁, which will in general be q-numbers but may be c-numbers.

Mathematically, this (together with some natural identities) means that observables form an algebraic structure known as a ring. Furthermore, the linear superposition principle imposes a liner structure upon observables, thus their set becomes an algebra. Some mathematically-oriented texts, e.g. [94]*§ 1.2, directly speak about an “algebra of observables” which is not far from the above quote [81]. It is also deducible from two connected statements in Dirac’s canonical textbook:

1. “the linear operators corresponds to the dynamical variables at that time” [82]*§ 7, p. 26.
2. “Linear operators can be added together” [82]*§ 7, p. 23.

However, the assumption that any two observables may be added cannot fit into a physical theory. To admit addition, observables need to have the same dimensionality. In the simplest example of the observables of coordinate q and momentum p, which units shall be assigned to the expression q+p? Meters or kilos×meters/seconds? If we get the value 5 for p+q in the metric units, what is then the result in the imperial ones? Since these questions cannot be answered, the above Dirac’s assumption is not a part of any physical theory.

Another common definition suffering from the same problem is used in many excellent books written by distinguished mathematicians, see for example [248]*§ 2-2 [104]*§ 1.1. It declares that quantum observables are projection-valued Borel measures on the dimensionless real line. Such a definition permit an instant construction (through the functional calculus) of new observables, including algebraically formed [248]*§ 2-2, p. 63:

Because of Axiom III, expressions such as A², A³+A, 1−A, and e^A all make sense whenever A is an observable.

However, if A has a physical dimension (is not a scalar) then the expression A³+A cannot be assigned a dimension in a consistent manner.¹

Of course, physical defects of the above (otherwise perfect) mathematical constructions do not prevent physicists from making correct calculations, which are in a good agreement with experiments. We are not going to analyse methods which allow researchers to escape the indicated dangers. Instead, it will be more beneficial to outline alternative mathematical foundations of quantum theory, which do not have those shortcomings.

1.3 Non-Essential Noncommutativity

While we can add two observables if they have the same dimension only, physics allows us to multiply any observables freely. Of course, the dimensionality of a product is the product of dimensionalities, thus the commutator [A,B]=AB−BA is well defined for any two observables A and B. In particular, the commutator (1) is also well-defined, but is it indeed so important?

In fact, it is easy to argue that noncommutativity of observables is not an essential prerequisite for quantum mechanics: there are constructions of quantum theory which do not relay on it at all. The most prominent example is the Feynman path integral. To focus on the really cardinal moments, we firstly take the popular lectures [97], which present the main elements in a very enlightening way. Feynman managed to tell the fundamental features of quantum electrodynamics without any reference to (non-)commutativity: the entire text does not mention it anywhere.

Is this an artefact of the popular nature of these lecture? Take the academic presentation of path integral technique given in [98]. It mentioned (non-)com­mu­ta­ti­vi­ty only on pages 115–6 and 176. In addition, page 355 contains a remark on noncommutativity of quaternions, which is irrelevant to our topic. Moreover, page 176 highlights that noncommutativity of quantum observables is a consequence of the path integral formalism rather than an indispensable axiom.

But what is the mathematical source of quantum theory if noncommutativity is not? The vivid presentation in [97] uses stopwatch with a single hand to explain the calculation of path integrals. The angle of stopwatch’s hand presents the phase for a path x(t) between two points in the configuration space. The mathematical expression for the path’s phase is [98]*(2-15):

where S[x(t)] is the classic action along the path x(t). Summing up contributions (2) along all paths between two points a and b we obtain the amplitude K(a,b). This amplitude presents very accurate description of many quantum phenomena. Therefore, expression (2) is also a strong contestant for the rôle of the cornerstone of quantum theory.

Is there anything common between two “principal” identities (1) and (2)? Seemingly, not. A more attentive reader may say that there are only two common elements there (in order of believed significance):

1. The non-zero Planck constant ℏ.
2. The imaginary unit i.

The Planck constant was the first manifestation of quantum (discrete) behaviour and it is at the heart of the whole theory. In contrast, classical mechanics is oftenly obtained as a semiclassical limit ℏ → 0. Thus, the non-zero Planck constant looks like a clear marker of quantum world in its opposition to the classical one. Regrettably, there is a common practice to “chose our units such that ℏ=1”. Thus, the Planck constant becomes oftenly invisible in many formulae even being implicitly present there. Note also, that 1 in the identity ℏ=1 is not a scalar but a physical quantity with the dimensionality of the action. Thus, the simple omission of the Planck constant invalidates dimensionalities of physical equations.

The complex imaginary unit is also a mandatory element of quantum mechanics in all its possible formulations. It is enough to point out that the popular lectures [97] managed to avoid any mention of noncommutativity but did uses complex numbers both explicitly (see the Index there) and implicitly (as rotations of the hand of a stopwatch). However, it is a common perception that complex numbers are a useful but manly technical tool in quantum theory.

1.4 Quantum Mechanics from the Heisenberg Group

Looking for a source of quantum theory we again return to the Heisenberg commutation relations (1): they are an important part of quantum mechanics (either as a prerequisite or as a consequence). It was observed for a long time that these relations are a representation of the structural identities of the Lie algebra of the Heisenberg group [104, 138, 139]. In the simplest case of one dimension, the Heisenberg group ℍ=ℍ¹ can be realised by the Euclidean space ℝ³ with the group law:

where ω is the symplectic form on ℝ² [11]*§ 37, which is behind the entire classical Hamiltonian formalism:

Here, like for the path integral, we see another example of a quantum notion being defined through a classical object.

The Heisenberg group is noncommutative since ω(x,y;x′,y′) =−ω(x′,y′;x,y). The collection of points (s,0,0) forms the centre of ℍ, that is (s,0,0) commutes with any other element of the group. We are interested in the unitary irreducible representations (UIRs) of ℍ in an infinite-dimensional Hilbert space H, that is a group homomorphism ((g₁)(g₂)=(g₁*g₂)) from ℍ to unitary operators on H. By Schur’s lemma, for such a representation , the action of the centre shall be multiplication by an unimodular complex number, i.e. (s,0,0)= e^{2πiℏ s} I for some real ℏ≠ 0.

Furthermore, the celebrated Stone–von Neumann theorem [104]*§ 1.5 tells that all UIRs of ℍ in complex Hilbert spaces with the same value of ℏ are unitary equivalent. In particular, this implies that any realisation of quantum mechanics, e.g. the Schrödinger wave mechanics, which provides the commutation relations (1) shall be unitary equivalent to the Heisenberg matrix mechanics based on these relations.

In particular, any UIR of ℍ is equivalent to a subrepresentation of the following representation on L₂(ℝ²):

Here ℝ² has the physical meaning of the classical phase space with q representing the coordinate in the configurational space and p—the respective momentum. The function f(q,p) in (5) presents a state of the physical system as an amplitude over the phase space. Thus, the action (5) is more intuitive and has many technical advantages [139, 341, 104] in comparison with the well-known Schrödinger representation (cf. (16)), to which it is unitary equivalent, of course.

Infinitesimal generators of the one-parameter subgroups ℏ(0,x,0) and ℏ(0,0,y) from (5) are the operators 1/2ℏ∂_p−2πi q and −1/2ℏ∂_q−2πi p. For these, we can directly verify the commutator identity:

Since we have a representation of (1), these operators can be used as a model of the quantum coordinate and momentum.

For a Hamiltonian H(q,p) we can integrate the representation ℏ with the Fourier transform Ĥ(x,y) of H(q,p):

and obtain (possibly unbounded) operator H′ on L₂(ℝ²). This assignment of the operator H′ (quantum observable) to a function H(q,p) (classical observable) is known as the Weyl quantization or a Weyl calculus [104]*§ 2.1. The Hamiltonian H′ defines the dynamics of a quantum observable k′ by the Heisenberg equation:

This is sketch of the well-known construction of quantum mechanics from infinite-di­men­sional UIRs of the Heisenberg group, which can be found in numerous sources [181, 104, 139].

1.5 Classical Noncommutativity

Now we are going to show that the priority of importance in quantum theory shall be shifted from the Planck constant towards the imaginary unit. Namely, we describe a model of classical mechanics with a non-zero Planck constant but with a different hypercomplex unit. Instead of the imaginary unit with the property i²=−1 we will use the nilpotent unit ε such that ε²=0. The dual numbers generated by nilpotent unit were already known for there connections with Galilean relativity [339, 115] – the fundamental symmetry of classical mechanics – thus its appearance in our discussion shall not be very surprising after all. Rather, we may wander why the following construction was unnoticed for such a long time.

Another important feature of our scheme is that the classical mechanics is presented by a noncommutative model. Therefore, it will be a refutation of Dirac’s claim about the exclusive rôle of noncommutativity for quantum theory. Moreover, the model is developed from the same Heisenberg group, which were used above to describe the quantum mechanics.

Consider a four-dimensional algebra C spanned by 1, i, ε and iε. We can define the following representation εh of the Heisenberg group in a space of C-valued smooth functions [199, 197]:

A simple calculation shows the representation property

for the multiplication (3) on ℍ. Since this is not a unitary representation in a complex-valued Hilbert space its existence does not contradict the Stone–von Neumann theorem. Both representations (5) and (8) are noncommutative and act on functions over the phase space. The important distinction is:

• The representation (5) is induced (in the sense of Mackey [159]*§ 13.4) by the complex-valued unitary character ℏ(s,0,0)= e^{2πiℏ s} of the centre of ℍ.
• The representation (8) is similarly induced by the dual number-valued character εh(s,0,0)= e^{εh s}=1+εh s of the centre of ℍ, cf. [194]. Here dual numbers are the associative and commutative two-dimensional algebra spanned by 1 and ε.

Similarity between (5) and (8) is even more striking if (8) is written² as:

Here, for a differentiable function k of a real variable t, the expression k(t+ε a) is understood as k(t)+ε a k′(t), where a∈ℂ is a constant. For a real-analytic function k this can be justified through its Taylor’s expansion, see [58] [342]*§ I.2(10) [115] [78] [79]. The same expression appears within the non-standard analysis based on the idempotent unit ε [30].

The infinitesimal generators of one-parameter subgroups εh(0,x,0) and εh(0,0,y) in (8) are

respectively. We calculate their commutator:

It is similar to the Heisenberg relation (1): the commutator is non-zero and is proportional to the Planck constant. The only difference is the replacement of the imaginary unit by the nilpotent one. The radical nature of this change becomes clear if we integrate this representation with the Fourier transform Ĥ(x,y) of a Hamiltonian function H(q,p):

This is a first order differential operator on the phase space. It generates a dynamics of a classical observable k – a smooth real-valued function on the phase space – through the equation isomorphic to the Heisenberg equation (7):

Making a substitution from (11) and using the identity ε²=0 we obtain:

This is, of course, the Hamilton equation of classical mechanics based on the Poisson bracket. Dirac suggested, see the paper’s epigraph, that the commutator corresponds to the Poisson bracket. However, the commutator in the representation (8) is exactly the Poisson bracket.

Note also, that both the Planck constant and the nilpotent unit disappeared from (12), however we did use the fact h≠ 0 to make this cancellation. Also, the shy disappearance of the nilpotent unit ε at the very last minute can explain why its rôle remain unnoticed for a long time.

1.6 Summary

We revised mathematical foundations of quantum and classical mechanics and the rôle of hypercomplex units i²=−1 and ε²=0 there. To make the consideration complete, one may wish to consider the third logical possibility of the hyperbolic unit є with the property є²=1 [142, 156, 199, 327, 281, 198, 194], see Section 4.4.

The above discussion provides the following observations [200]:

1. Noncommutativity is not a crucial prerequisite for quantum theory, it can be obtained as a consequence of other fundamental assumptions.
2. Noncommutativity is not a distinguished feature of quantum theory, there are noncommutative formulations of classical mechanics as well.
3. The non-zero Planck constant is compatible with classical mechanics. Thus, there is no a necessity to consider the semiclassical limit ℏ → 0, where the constant has to tend to zero.
4. There is no a necessity to request that physical observables form an algebra, which is a physical non-sense since we cannot add two observables of different dimensionalities. Quantization can be performed by the Weyl recipe, which requires only a structure of a linear space in the collection of all observables with the same physical dimensionality.
5. It is the imaginary unit in (1), which is ultimately responsible for most of quantum effects. Classical mechanics can be obtained from the similar commutator relation (10) using the nilpotent unit ε²=0.

In Dirac’s opinion, quantum noncommutativity was so important because it guaranties a non-trivial commutator, which is required to substitute the Poisson bracket. In our model, multiplication of classical observables is also non-commutative and the Poisson bracket exactly is the commutator. Thus, these elements do not separate quantum and classical models anymore.

Our consideration illustrates the following statement on the exceptional rôle of the complex numbers in quantum theory:

…for the first time, the complex field ℂ was brought into physics at a fundamental and universal level, not just as a useful or elegant device, as had often been the case earlier for many applications of complex numbers to physics, but at the very basis of physical law. [277]

In the 1960s it was said (in a certain connection) that the most important discovery of recent years in physics was the complex numbers. [250]*p. 90

Thus, Dirac may be right that we need to change a single assumption to get a transition between classical mechanics and quantum. But, it shall not be a move from commutative to noncommutative. Instead, we need to replace a representation of the Heisenberg group induced from a dual number-valued character by the representation induced by a complex-valued character. Our conclusion can be stated like a proportionality:

Classical/Quantum=Dual numbers/Complex numbers.

1

These arguments are rooted in history. Vieta, the founding father of the symbolic algebra, always wrote x²+x· 1 instead of x²+x: the former avoids “meaningless addition of an area to a length” expressed by the later. However, very soon Descartes said that Vieta’s notations used unnecessary geometric justifications.

2

I am grateful to Prof. N.A.Gromov, who suggested this expression.