All posts by ajb

What is geometry?

This is a question I am not really sure how to answer. So I put it to Sir Michael Atiyah after his Frontiers talk in Cardiff. In essence he told me that geometry is any mathematics that you can imagine as pictures in your head.

To me this is in fact a very satisfactory answer. Geometry a word that literally means “Earth Measurement” has developed far beyond its roots of measuring distances, examining solid shapes and the axioms of Euclid.

Another definition of geometry would be the study of spaces. Then we are left with the question of what is a space?

Classically, one thinks of spaces, say topological or vector spaces as sets of points with some other properties put upon them. The notion of a point seems deeply tied into the definition on a space.

This is actually not the case. For example all the information of a topological space is contained in the continuous functions on that space. Similar statements hold differentiable manifolds for example. Everything here is encoded in the smooth functions on a manifold.

This all started with the Gelfand representation theorem of C*-algebras, which states that “commutative C*-algebras are dual to locally compact Hausdorff spaces”. I won’t say anything about C*-algebras right now.

In short one can instead of studying the space itself one can study the functions on that space. More than this, one can take the attitude that the functions define the space. In this way you can think of the points as being a derived notion and not a fundamental one.

This then opens up the possibility of non-commutative geometries by thinking of non-commutative algebras as “if they were” the algebra of functions on some non-commutative space.

Also, there are other constructions found in algebraic geometry that are not set-theoretical. Ringed spaces and schemes for example.

So, back to the opening question. Geometry seems more like a way of thinking about problems and constructions in mathematics rather than a “stand-alone” topic. Though the way I would rather put it that all mathematics is really geometry!

Should you beleive everything on the arXiv?

For those of you who do not know, the arXiv is an online repository of reprints in physics, mathematics, nonlinear science, computer science, qualitative biology, qualitative finance and statistics. In essence it is a place that scientists can share their work and work in progress, but note that it is not peer reviewed. The arXiv is owned and operated by Cornell University and all submissions should be in line with their academic standards.

So, can you believe everything on the arXiv?

In my opinion overall the arXiv is contains good material and is a vital resource for scientists to call upon. Many new works can be made public this way, before being published in a scientific journal. Indeed, most of the published papers I have had call to use have versions on the arXiv. Moreover, the service is free and requires no subscription.

However, there can be errors and mistakes in the preprints, both “editorial” but more importantly scientifically. Interestingly, overall the arXiv is not full of crackpot ideas despite it being quite open. There is a system of endorsement in place meaning that an established scientist should say that the first preprint you place on the arXiv is of general interest to the community. This stops the very eccentric quacks in their tracks.

There has been some widely publicised examples of preprints on the arXiv that have cursed a stir within the scientific community. Two well-known examples include

A. Garrett Lisi, An Exceptionally Simple Theory of Everything arXiv:0711.0770v1 [hep-th],

and more recently

V.G.Gurzadyan and R.Penrose, Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity arXiv:1011.3706v1 [astro-ph.CO],

both of which have received a lot of negative criticism. Neither has to date been published in a scientific journal.

Minor errors and editing artefacts can be corrected in updated versions of the preprints. Should preprints on the arXiv be found to be in grave error, the author can withdraw the preprint.

With that in mind, the arXiv can be a great place to generate feedback on your work. I have done this quite successfully in the past. This allowed me to get some useful comments and suggestion on work, errors and all.

My advice is to view all papers and preprints with some scepticism, even full peer review can not rule out errors. Though, always be more confident with published papers and arXiv preprints that have gone under some revision. Note that generally people who place preprints on the arXiv are not trying to con or trick anyone, all errors will be genuine mistakes.

Odd Jacobi structures and BV-gauge systems

In this paper we define Grassmann odd analogues of Jacobi structures on supermanifolds. We then examine their potential use in the Batalin-Vilkovisky formalism of classical gauge theories.

arXiv:1101.1844v1 [math-ph]

In my latest preprint I construct a Grassmann odd analogue of Jacobi structures on supermanifolds.

Without any details (being slack with signs) an odd Jacobi structure on a supermanifold is an ” almost Schouten structure”, \(S\) that is an odd function on the total space of the cotangent bundle of the supermanifold quadratic in fibre coordinates and a homological vector field \(Q\) on the supermanifold together with the compatibility conditions

\(L_{Q}S = \{\mathcal{Q}, S \} = 0\),
\( \{S, S \} = 2 S \mathcal{Q}\),

where \(\mathcal{Q} \in C^{\infty}(T^{*}M)\) is the “Hamiltonian” or principle symbol of the Homological vector field. The brackets here are the canonical Poisson brackets on the cotangent bundle.

An odd Lie bracket can then be constructed on \(C^{\infty}(M)\)

\([f,g] = \pm \{ \{ S,f \},g\} \pm \{ \mathcal{Q},fg \} \).

So, this odd bracket satisfies all the properties of a Schouten bracket i.e. symmetry and the Jacobi identity, but the Leibniz rule is not identically satisfied. There is an “anomaly” to the Leibniz rule of the form

\( [f,gh] = \pm [f,g]h \pm g [f,h] \pm [f,1] gh\)

In the preprint I examine the basic properties of odd Jacobi manifolds. The definition and study of odd Jacobi manifolds appears to be missing from the previous literature despite the wide interest in Schouten manifolds and Q-manifolds in mathematical physics.

One should note that for classical or even Jacobi structures (if you know what these are) the Reeb vector field has no constrain on it like being homological. For odd structures the homological condition is essential.

I also consider if the classical BV-antifield formalism can be generalised to odd Jacobi manifolds. In short, does one require the antibracket to be a Schouten bracket or can one weaken the Leibniz rule? I show that it looks possible to extend the BV formalism, classically anyway to odd Jacobi manifolds with the extra condition that the extended classical action not just be a Maurer-Cartan element,

\([s,s] = 0\),

but in addition should be Q-closed,

\(Qs =0\).

Much work needs to be done to generalise the BV formalism to odd Jacobi manifolds including adding the required gradings of ghost number, antifield number etc as well as understanding the quantum aspects.

UPDATE: 22 March 2011. I have found a mistake in one of the examples I suggest. This is corrected and an updated version of the preprint will appear in due course. The mistake does not really effect the rest of the preprint.

Integration of odd variables III

We will proceed to describe how changes of variables effects the integration measure for odd variables. We will do this via a simple example rather than in full generality.

Integration measure with two odd variables
Let us consider the integration with respect to two odd variables, \(\{ \theta, \overline{\theta} \}\). Let us consider a change in variables of the form

\(\theta^{\prime} = a \theta + b \overline{\theta}\),
\( \overline{\theta}^{\prime} = c \theta + d \overline{\theta}\),

where a,b,c,d are real numbers (or complex if you wish).

Now, one of the basic properties of integration is that it should not depend on how you parametrise things. In other worlds we get the same result whatever variables we chose to employ. For the example at hand we have

\( \int D(\overline{\theta}^{\prime}, \theta^{\prime}) \theta^{\prime} \overline{\theta}^{\prime} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

Thus, we have

\(\int D(\overline{\theta}^{\prime}, \theta^{\prime}) (ad-bc)\theta \overline{\theta} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

In order to be invariant we must have

\(\int D(\overline{\theta}^{\prime}, \theta^{\prime})= \frac{1}{(ad-bc) }D(\overline{\theta}, \theta) \).

Note that the factor (ad-bc) is the determinant of a 2×2 matrix. However, note that we divide by this factor and not multiply in the above law. This is a general feature of integration with respect to odd variables, one divides by the determinant of the transformation matrix rather than multiply. This generalises to non-linear transformations that mix even and odd coordinates on a supermanifold. This is the famous Berezinian. A detailed discussion is outside the remit of this introduction.

Furthermore, note that the transformation law for the measure is really the same as the transformation law for derivatives. Thus, the Berezin measure is really a mixture of algebraic and differential ideas.

What next?
I think this should end our discussion of the elementary properties of analysis with odd variables. I hope it has been useful to someone!

Integration of odd variables II

We now proceed to define integration with respect to odd variables.

The fundamental theorem of calculus for odd variables
Let us consider just one odd variable. This will be sufficient for our purposes for now. Following the direct analogy with integration of functions over a circle the fundamental theorem of calculus states

\(\int D\theta \frac{\partial f(\theta)}{\partial \theta} =0\).

We use the notation \(D\theta \) for the measure rather than \(d\theta \) as the measure cannot be associated with a one-form. We will discuss this in more detail another time.

Definition of integration
Recall that the general form of a function in one odd variable is

\(f(\theta) = a + \theta b\),

with a and b being real numbers. Thus from the fundamental theorem we have

\(\int D\theta b =0\).

In particular this implies

\(\int D\theta =0\).

Then we have

\(\int D\theta f(\theta) = a \int D\theta + b \int D\theta \:\: \theta = b \int D\theta\:\: \theta \).

Thus to define integration all we have to do is define the normalisation

\(\int D\theta\:\: \theta\).

The choice made by Berezin was to set this to unity. Other choices are also just as valid. Thus,

\(\int D\theta f(\theta) = b\).

Integration for several odd variables
For the case of more than one odd variable one simply uses

\(\int D(\theta_{1}, \theta_{2} , \cdots \theta_{n})f(\theta) = \int D\theta_{1} \int D \theta_{2} \cdots \int D\theta_{n} f(\theta)\).

example Consider two odd variables.

\(\int D(\overline{\theta}, \theta) \left( f_{0} + \theta \:f + \overline{\theta}\: \overline{f} + \theta \overline{\theta}F \right) = F \).

The general rule is that (taking care with signs) the integration with respect to the measure \(D(\theta_{1}, \theta_{2} , \cdots \theta_{n})\) of a function picks out the coefficient of the \(\theta_{1}, \theta_{2} , \cdots \theta_{n}\) term.

Integration and differentiation are the same!
From the above we see that differentiation with respect to an odd variable is the same as integration with respect to the odd variable. This explains why we cannot associate a “top-form” with the measure. This will become more apparent when we discuss changes of variables.

What next?
Next we will examine how changing variables in the integration effects the measure. We will see that things look “upside down” as compared with the integration of real variables. This is anticipated by the equivalence of integration and differentiation.

Integration of odd variables I

Before we consider odd variables, let us describe how to algebraically define integration of functions over the circle.

Functions on the circle
Recall the Fourier expansion. It is well known that any continuous function on the circle is of the form

\(f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right) \),

with the a’s and b’s being constants, i.e. independent of the variable x.

The fundamental theorem of calculus
The fundamental theorem of calculus states that

\(\int_{S^{1}} dx \: \frac{\partial f(x)}{\partial x } = 0 \),

as functions on the circle are periodic.

Integration of functions
It turns out that integration of functions over the circle can be defined algebraically up to a choice in measure. To see this observe

\(\int_{S^{1}} dx f(x) = \int_{S^{1}} dx \frac{a_{0}}{2} + \int_{S^{1}} dx \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right)\)

Then we can write

\(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx + \int_{S_{1}} dx \frac{\partial }{\partial x} \sum_{n=1}^{\infty} \left ( \frac{a_{n}}{n}\sin(nx) + \frac{- b_{n}}{n} \cos(nx) \right)\)

to get via the fundamental theorem of calculus

\(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx\).

So we have just about defined integration completely algebraically from the fundamental theorem of calculus. All we have to do is specify the normalisation

\(\int_{S^{1}} dx \).

The standard choice would be

\(\int_{S^{1}} dx = 2 \pi\),

to get back to our usual notion of integration of periodic functions. Though it would be quite consistent to consider some other normalisation, say to unity.

Anyway, up to a normalisation the integration of functions over the circle selects the “constant term” of the corresponding Fourier expansion.

What next?
So, the above construction demonstrates that integration of functions over a domain without boundaries can be defined algebraically, up to a normalisation. This served as the basis for Berezin who defined the notion of integration of odd variables.

Recall that odd variables have no topology and no boundaries. The integration with respect to such variables cannot be in the sense of Riemann. However, thinking of functions of odd variables in analogy to periodic functions integration can be defined algebraically. We will describe this next time.

Differential calculus of odd variables.

Here we will define the notion of differentiation with respect to an odd variable and examine some basic properties.

Differentiation with respect to an odd variable is completely and uniquely defined via the following rules:

  1. \(\frac{ \partial \theta^{\beta} }{\partial \theta^{\alpha}} = \delta_{\alpha}^{\beta} \).
  2. Linearity:
    \(\frac{\partial}{\partial \theta }(a f(\theta)) = a \frac{\partial}{\partial \theta } f(\theta)\).
    \(\frac{\partial}{\partial \theta }( f(\theta) + g(\theta)) = \frac{\partial}{\partial \theta }f(\theta) + \frac{\partial}{\partial \theta } g(\theta)\).
  3. Leibniz rule:
    \(\frac{\partial}{\partial \theta }( f(\theta)g(\theta)) = \frac{\partial f(\theta)}{\partial \theta } + (-1)^{\widetilde{f}} f \frac{\partial g(\theta)}{\partial \theta } \).

The operator \(\frac{\partial }{\partial \theta }\) is odd, that is it changes the parity of the function it acts on. This must be taken care of when applying Leibniz’s rule.

Elementary properties
It is easy to see that

\(\frac{\partial}{\partial \theta^{\alpha}}\frac{\partial}{\partial \theta^{\beta}}+ \frac{\partial}{\partial \theta^{\beta}}\frac{\partial}{\partial \theta^{\alpha}}=0\),

in particular

\(\left( \frac{\partial}{\partial \theta} \right)^{2}=0\).

\(\frac{\partial}{\partial \theta} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = b + \overline{\theta}d\).

\(\frac{\partial}{\partial \overline{\theta}} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = c- \theta d\).

Changes of variables
Under changes of variable of the form \(\theta \rightarrow \theta^{\prime}\) the derivative transforms as standard

\(\frac{\partial}{\partial \theta^{\prime}} = \frac{\partial\theta}{\partial \theta^{\prime}} \frac{\partial}{ \partial \theta}\).

We will have a lot more to say about changes of variables (coordinates) another time.

What next?
We now know how to define and use the derivative with respect to an odd variable. Note that this was done algebraically with no mention of limits. As the functions in odd variables are polynomial the derivative was simple to define.

Next we will take a look at integration with respect to an odd variable. We cannot think in terms of boundaries, limits or anything resembling the Riemann or Lebesgue notions of integration. Everything will need to be done algebraically.

This will lead us to the Berezin integral which has the strange property that integration and differentiation with respect to an odd variable are the same.

Elementary algebraic properties of superalgebras

Here we will present the very basic ideas of Grassmann variables and polynomials over them.

Grassmann algebra
Consider a set of n odd variables \(\{ \theta^{1}, \theta^{2}, \cdots \theta^{n} \}\). By odd we will mean that they satisfy

\( \theta^{a}\theta^{b} + \theta^{b} \theta^{a}=0\).

Note that in particular this means \(\theta^{2}=0\). That is the generators are nilpotent.

The Grassmann algebra is then defined as the polynomial algebra in these variables. Thus a general function in odd variables is

\(f(\theta) = f_{0} + \theta^{a}f_{a} + \frac{1}{2!} \theta^{a} \theta^{b}f_{ba} + \cdots + \frac{1}{n!} \theta^{a_{1}} \cdots \theta^{a_{n}}f_{a_{n}\cdots a_{1}}\).

The coefficients we take as real and antisymmetric. Note that the nilpotent property of the odd variables means that the Grassmann algebra is complete as polynomials.

Example If we have the algebra generated by a single odd variable \(\theta \) then polynomials are of the form

\(a + \theta b\).

Example If we have two odd variables \(\theta\) and \(\overline{\theta}\) then polynomials are of the form

\(a + \theta b + \overline{\theta} c + \theta \overline{\theta} d\).

It is quite clear that the polynomials in odd variables forms a vector space. You can add such functions and multiply by a real number and the result remains a polynomial. It is also straightforward to see that we have an algebra. One can multiply two such functions together and get another.

The space of all such functions has a natural \(\mathbb{Z}_{2}\)-grading, which we will call parity given by the number of odd generators in each function mod 2. If the function has an even/odd number of odd variables then the function is even/odd. We will denote the parity by of a function \(\widetilde{f}= 0/1\), if it is even/odd.

Example \(a +\theta \overline{\theta} d \) is an even function and \(\theta b + \overline{\theta} c \) is an odd function.

Let us define the (super)commutator of such functions as

\([f,g] = fg -(-1)^{\widetilde{f} \widetilde{g}} gf\).

If the functions are not homogeneous, that is even or odd the commutator is extended via linearity. We see that the commutator of any two functions in odd variables vanishes. Thus we say that the algebra of functions in odd variables forms a supercommutative algebra.

Specifically note that this means the ordering of odd functions is important.

The modern approach to geometry is to define and deal with “spaces” in terms of the functions upon them. Geometrically we can think of the algebra generated by n odd variables as defining the space \(\mathbb{R}^{0|n}\). Note that no such “space” in the classical sense exists. In fact such spaces consist of only one point!

If we promote the coefficients in the polynomials to be functions of m real variables then we have the space \(\mathbb{R}^{m|n}\). We are now most of the way to defining supermanifolds, but this would be a digression from the current issues.

Noncommutative superalgebras
Of course superalgebras for which the commutator generally is non-vanishing can be defined and are naturally occurring. We will encounter such things when dealing with first order differential operators acting on functions in odd variables. Geometrically these are the vector fields. Recall that the Lie bracket between vector fields over a manifold is in general non-vanishing.

What next?
Given the basic algebraic properties of functions in odd variables we will proceed to algebraically define how to differentiate with respect to odd variables.

Introduction to Superanalysis

Following a conversation on a popular science chat room the subject of Grassmann variables and in particular the Berezin integral arose. Thus I decided to with a short introduction to the basic theory of superalgebras, particularly supercommutative algebras and their calculus.

We will be primarily interested in algebras that involve the Koszul sign rule, that is include an extra minus sign when you interchange odd elements:

\(ab = – ba\).

Ancient History
The beginning of all supermathematics can be traced back to 1885 and the work of Hermann Günther Grassmann on linear algebra. He introduced variables that involve a minus sign when interchanging their order. Élie Cartan’s theory of differential forms is also in hindsight a “super-theory”. Many other constructions in algebra and topology can be thought of as “super” and involve a sign factor when interchanging the order.

By the early 1950’s odd variables appeared in quantum field theory as a semiclassical description of fermions. Initially the analysis was based on the canonical description of quantisation and so confined to derivatives with respect to odd variables. Berezin in 1961 introduced the integration theory for odd variables and this was promptly applied to the path integral approach to quantisation.

In these early works odd variables were understood very formally in an algebraic way. That is they were not associated with with any general notion of a space. Berezin’s treatment of even and odd variables convinced him that there should be a way to treat them analogously to real and complex variables in complex geometry. The bulk of this work was carried out by Berezin and his collaborators between 1965 and 1975. Berezin introduced general non-linear transformations that mix even and odd variables as well as generalisation of the determinant to integration over even and odd variables. This work led to the notion of superspaces and supermanifolds. In essence one thinks of a supermanifold as a “manifold” with even (commuting) and odd (anticommuting) coordinates. A detailed discussion of supermanifolds is out of the scope of this introduction.

Supersymmetric field theories
The nomenclature super comes from physics. Gol’fand & Likhtman extended the Poincare group to include “odd translations”. These operators are fermionic in nature and thus require anticommutators in the extended Poincare algebra. Supersymmetry is a remarkable symmetry that mixed bosonic and fermionic degrees of freedom. Lagrangians (or actions) that exhibit supersymmetry have some very attractive features. The surprising result is that supersymmetry can cancel most or even all of the divergences of certain quantum field theories. A detailed discussion of supersymmetric field theories is outside the scope of this introduction.

Gauge theories and the BRST symmetry
The use of odd variables is also necessary in (perturbative) non-abelian gauge theories (in the covariant gauges at least), even if one initially restricts attention to theories without fermions. There are several complications that do not arise in abelian gauge theory. These originate primarily from the gauge fixing, which effects the path integration measure in a non-trivial way. Feynman in 1963 showed that using standard quantisation methods available at the time, Yang-Mills theory was not unitary. Feynman also showed that counter terms, now known as ghosts could be added that remove the nonunitary parts. Originally these ghost, which are odd but violate the spin-statistics theorem were seen as ad-hoc. Later Faddeev and Popov showed that these ghost arise in the theory by considering the so called Faddeev-Popov determinant.

It was noticed that the gauge fixed Lagrangian possess a new global (super)symmetry that rotates the gauge fields into ghosts. This symmetry is named after it’s discoverers Becchi, Rouet, Stora and independently Tyutin, thus BRST symmetry. As this is a global symmetry no new degrees of freedom can be eliminated.

The BRST symmetry is now a fundamental tool when dealing with quantum gauge theories. For example the BRST symmtery is important when considering the remormalisability and absence of anomalies for a given theory. We will not say any more about gauge theories in this introduction.

Mathematical applications
Odd elements can be employed very successfully in pure mathematics. For example, the de Rham complex of a manifold can be completely understood in terms of functions and vector fields over a particular supermanifold. Multivector fields can also be thought of in a similar way in terms of a supermanifold and an odd analogue of a Poisson bracket.

Various algebraic structures can be encoded in superalgebras that come equipped with a homological vector field. That is an odd vector field that “squares to zero”

\(Q^{2} = \frac{1}{2}[Q,Q]=0\).

Common examples include Lie algebras, \(L_{\infty}\)-algebras, Lie algebroids, \(A_{\infty}\)-algebra etc.

Guide to this introduction
I hope that these opening words have convinced you that the study of superalgebras and Grassmann odd variables is useful in physics and pure mathematics.

I will be quite informal in presentation and attitude. The intention is to convey the main ideas without over burdening the reader.

A tentative guide is as follows:

  1. Elementary algebraic properties of superalgebras.
  2. Differential calculus of odd variables.
  3. Integration with respect to odd variables: the Berezin integral.

Quick guide to references
The mathematical theory of Grassmann algebras, superalgebras and supermanifolds is well established and can be found in several books. Any book on quantum field theory will say something about the algebra and calculus of odd variables. The mathematical books that I like include:

  • Gauge Field Theory and Complex Geometry, Yuri I. Manin, Springer; 2nd edition (June 27, 1997).
  • Geometric Integration Theory on Supermanifolds, Th. Th. Voronov, Routledge; 1 edition (January 1, 1991).
  • Supersymmetry for Mathematicians: An Introduction, V. S. Varadarajan, American Mathematical Society (July 2004).

Other books that deserve a mention are

  • Supermanifolds, Bryce DeWitt, Cambridge University Press; 2 edition (June 26, 1992).
  • Supermanifolds: Theory and Applications, A. Rogers, World Scientific Publishing Company (April 18, 2007).

Quantum Algebra?

“Quantum algebra” is used as one top-level mathematics categories on the arXiv. However, to me at least it is not very clear what is meant by the term.

Topics in this section include

  • Quantum groups and noncommutative geometry
  • Poisson algebras and generalisations
  • Operads and algebras over them
  • Conformal and Topological QFT

Generally these include things that are not necessarily commutative.

What is a commutative algebra? Intentionally being very informal, an algebra is a vector space over the real or complex numbers (more generally any field) endowed with a product of two elements.

So let us fix some vector space \(\mathcal{A}\) say over the real numbers. It is an algebra if there is a notion of multiplication of two elements that is associative

\(a(bc) = (ab)c\)

and distributive

\(a(b+c) = ab + ac\),

with \(a,b,c \in \mathcal{A}\). There may also be a unit

\(ea = ae \) for all \(a \in \mathcal{A}\). Sometimes there may be no unit.

An algebra is commutative if the order of the multiplication does not matter. That is

\(ab = ba\).

For example, if \(a\) and \(b\) are real or complex numbers then the above holds. So real numbers and complex numbers can be thought of as “commutative algebras over themselves”.

It is common to define a commutator as

\([a,b] = ab – ba\).

If the commutator is zero then the algebra is commutative. If the commutator is non-zero then the algebra is noncommutative. In the second case the order of multiplication matters

\(ab \neq ba \)

in general.

The first example here is the algebra of 2×2 matrices.

So why “quantum”? Of course noncommutative algebras were known to mathematicians before the discovery of quantum mechanics. However, they were not generally known by physicists. The algebras used in classical mechanics, say in the Hamiltonian description are all commutative. Here the phase space is described by coordinates \(x,p \) or equivalently by the algebra of functions in these variables. This algebra is invariably commutative.

In quantum mechanics something quite remarkable happens. The phase space gets replaced by something noncommutative. We can think of “local coordinates” \(\hat{x}, \hat{p}\) that are no longer commutative. In fact we have

\([\hat{x}, \hat{p}] = i \hbar \),

which is known as the canonical commutation relation and is really the fundamental equation in quantum mechanics. The constant \(\hbar\) is known as Planck’s constant and sets the scale of quantum theory.

The point being that quantum mechanics means that one must consider noncommutative algebras. Thus the relatively informal bijection “quantum” \(\leftrightarrow \) “noncommutative”.

We can also begin to understand Einstein’s dislike of quantum mechanics, as pointed out by Dirac. The theories of special and general relativity are by their nature very geometric. As I have suggested, a space can be thought of as being defined by the algebra of functions on it. Einstein’s theories are based on commutative algebra. Quantum mechanics on the other hand is based on noncommutative algebra and in particular the phase space is some sort of “noncommutative space”. The thought of a noncommutative space, “the coordinates do not commute” should make you shudder the first time you hear this!

One place you should pause for reflection is the notion of a point. In noncommutative geometry there is no elementary intuitive notion of a point. Noncommutative geometry is pointless geometry!

We can understand this via the quantum mechanical phase space and the Heisenberg uncertainty relation. Recall that the uncertainty principle states that one cannot know simultaneously the position and momentum of a quantum particle. One cannot really “select a point” in the phase space. The best we have is

\(\delta \hat{x} \delta \hat{p} \approx \frac{\hbar}{2}\).

The phase space is cut up into fuzzy Bohr-Heisenberg cells and does not consist of a collection of points.

At first it seems that all geometric intuition is lost. This however is not the case if we think of a space in terms of the functions on it. A great deal of noncommutative geometry is rephrasing things in classical differential geometry in terms of the functions on the space (the structure sheaf). Then the notion may pass to the noncommutative world. I should say more on noncommutative geometry another time.