Fifteen Eighty Four

Academic perspectives from Cambridge University Press


Quantum Anomaly and the Origin of Time

Shahn Majid

First off, newly syndicated readers who want to have access to my previous posts can find them archived here as well as listed on my own site here.

After last week’s speculations on time I would like to ask an even deeper question: why is there time?

My 4 year old daughter would be proud. What I mean is, why do things evolve in the first place? It seems to me that fundamental physics has to answer not only ‘what’ questions but also ‘why’ questions if it claims to provide understanding. I think I have an answer, or a glimpse of one.


The answer has to do with quantum anomalies; no not the large (not very quantum, then) things that seem to turn up in every other episode of Star Trek Voyager, but what physicists mean by this, which I am afraid is much more dry and dusty. In fact, I’m going to have to ask you to dust off your high school calculus books, just for a minute.

I explained in a previous post that even if nobody at the moment knows how to reconcile quantum theory and gravity, quantum spacetime should emerge as an effect coming out of any unknown theory. Typically, the coordinates x,y,z of space would also be quantum variables, so space alone should typically form some kind of symbolic algebra. Due to quantum effects, the order of the variables in this algebra will matter, xy will typically not coincide with yx. One says that the algebra is ‘noncommutative’.

Click to enlarge

Now, what about differential calculus on such a quantum space? If you remember any high school calculus it means things like dx, dy, dz as the ‘infinitesimal differences’. Newton and Leibniz both considered such things as numbers which are then made arbitrarily small. Hands up if your high school calculus class contained a picture like the one shown at left. It defines differentiation of a function f in the x direction as a limit of the slope df/dx of the triangle as dx gets small.

So to develop quantum gravity effects in physics we also need ‘quantum differentials’ dx, dy, dz. They should enjoy the properties that differentials enjoy in Newtons theory except, since xy and yx need not coincide, similarly y dx need not coincide with dx y, etc. Now, here is the remarkable thing one finds as you dig deeper into this world of quantum geometry:

Most sufficiently noncommutative ‘quantum spaces’ described by algebras with variables x,y,z do not admit any reasonable self-contained algebra of differentials dx,dy,dz.

This is called an anomaly for quantum differentiation, or quantum anomaly for short. In physics when a classical symmetry does not survive the theoretical passage from our understanding at the level of classical mechanics to quantum mechanics, one speaks of an `anomaly’. An example of an anomaly is when physicists first tried string theory in the realistic four spacetime dimensions. Their theory had an anomaly for conformal symmetry and this was ‘fixed’ by changing the dimension to 24 and later to 11 or 10, though nowadays I hear that 10500 dimensions is currently quite popular. In that sense, if you subscribe to string theory, you have a ‘prediction’ or possible necessity of higher dimensions.

In my case what I mean by ‘reasonable’ is a differential calculus that respects in some form the symmetry that should rotate the x,y,z among themselves. Among such calculi there is an obstruction uncovered for many quantum spaces in my work with my colleague Edwin Beggs, a mathematician at Swansea in the UK.

As is typical for any anomaly, the thing to do is to increase the dimensions to absorb the obstruction. So if you subscribe to quantum spaces, you will typically be forced to invent at least one extra dimension which I will call ‘dt’. In fact you will be forced to invent time.

Let’s put some flesh on this. From the differentials dx, dy, dz, dt you can recover the corresponding ‘differentiation in direction x,y,z,t’ operations (the so-called partial derivatives), and you can do it algebraically without recourse to pictures. If you let f(x,y,z) be an element of our quantum space algebra, you find that its ‘differentiation in the t direction’ comes out not as zero but, in the simplest models, as something like the energy operator in Schroedinger’s equation.

So, not only are you forced to invent time, your original space variables obey an equation which in the limit of ordinary space (i.e. as you remove quantum gravity effects) becomes Schroedinger’s wave equation. Oops, you’ve just answered the question ‘why is there quantum mechanics?’ My 4 year old has not even gotten to that one.

And none of this is an accident. For any sufficiently ‘quantum’ space there are general mathematical reasons to expect that for any reasonable differential d there will exist an abstract differential element which I will call ‘dt’ obeying

dt f – f dt = L df

where L is our Planck scale parameter expressing the effects of quantum gravity. So, if t was not one of your variables you would be forced to invent it so as to have dt. Note that both sides of the equation are zero in ordinary geometry. In other words, this equation is invisible until you study quantum gravity, which is why such an origin of time is not seen in ordinary classical and quantum physics.

Editors note: Shahn Majid was talking around his paper “Noncommutative model with spontaneous time generation and Planckian-bound”, Journal of Mathematical Physics, 2005. A general introduction to his ideas on quantum spacetime appear in On Space and Time.

About The Author

Shahn Majid

Shahn Majid is co-author of On Space and Time (2008) and author of A Quantum Groups Primer (2002) and Foundations of Quantum Group Theory (2000). He is Professor of Mathematics at ...

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