Quantum Mechanics and the Fourier-Legendre Transform over a Semiring
Update: see also my more recent notes on this subject.
Consider the two semirings: (R,+,*) and (R',min,+) where R' is R with a positive infinity value added in. The former is just the real numbers added and multiplied in the usual way and in the latter we use the operations min (which gives the lower of its two arguments) and + instead. There are many formal similarities shared by these structures. For example we have the distributive laws a*(b+c) = a*b+a*c and a+min(b,c) = min(a+b,a+c). Zero acts as the identity for + and we have a+0 = a and infinity is the identity for min with min(a,infinity) = a.
It turns out that the former semiring can be viewed as a quantum version of the latter semiring. In particular we can frequently take statements from quantum mechanics and consider them to be statements in a more general semiring rather than over (R,+,*). When we interpret these more general statements in the semiring (R',min,+) they turn out to say things about classical mechanics.
Consider for example the Hamiltonian formulation of classical mechanics. This essentially says that dynamical systems evolve in such a way that the integral of the Lagrangian is minimised. In other words, the integral of the Lagrangian is the min of its value for all possible paths. In quantum mechanics we no longer have systems taking the minimum but in a sense they take all paths. To compute physical quantities we must instead use the Feynman path integral to integrate over all paths. The factor we must integrate is essentially the exponential of the Lagrangian. In classical mechanics we have the min over an infinite set, in quantum mechanics we have the sum (ie. integral) over the same set. See here for a recent paper on this subject.
Another surprising analogy between these two semirings arises we we try to transfer the concept of the Fourier transform to (R,+,*) to (R',min,+). It's not obvious how to interpret the exponential function in (R',min,+) but it turns out that the natural choice is to consider the ordinary linear functions (in the conventional sense) to be the correct analogue. If we then replace the exponentials with linear functions in the definition of the Fourier transform and replace the integral with an infinite min what we end up with is another familiar transform: the Legendre transform. So the Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings.
The analogy carries quite far. In classical mechanics the Legendre transform converts between the Lagrangian and Hamiltonian formulations of classical mechanics. So it changes equations of motion written in terms of (generalised) position into equations written in terms of momentum and vice versa. In quantum mechanics the Fourier transform does much the same thing: the Fourier transform of a wavefunction in space gives the wavefunction in momentum space. Sean Walston has a paper on this. I'm not sure he was aware of the semiring connection when he wrote that.
In summary we have:
I never did understand the Legendre transform. It always seemed like this strange thing plucked out of nowhere. So it's amazing to see that in some sense it is the 'right' thing to study and is as natural as the Fourier transform. Fascinating stuff!
Consider the two semirings: (R,+,*) and (R',min,+) where R' is R with a positive infinity value added in. The former is just the real numbers added and multiplied in the usual way and in the latter we use the operations min (which gives the lower of its two arguments) and + instead. There are many formal similarities shared by these structures. For example we have the distributive laws a*(b+c) = a*b+a*c and a+min(b,c) = min(a+b,a+c). Zero acts as the identity for + and we have a+0 = a and infinity is the identity for min with min(a,infinity) = a.
It turns out that the former semiring can be viewed as a quantum version of the latter semiring. In particular we can frequently take statements from quantum mechanics and consider them to be statements in a more general semiring rather than over (R,+,*). When we interpret these more general statements in the semiring (R',min,+) they turn out to say things about classical mechanics.
Consider for example the Hamiltonian formulation of classical mechanics. This essentially says that dynamical systems evolve in such a way that the integral of the Lagrangian is minimised. In other words, the integral of the Lagrangian is the min of its value for all possible paths. In quantum mechanics we no longer have systems taking the minimum but in a sense they take all paths. To compute physical quantities we must instead use the Feynman path integral to integrate over all paths. The factor we must integrate is essentially the exponential of the Lagrangian. In classical mechanics we have the min over an infinite set, in quantum mechanics we have the sum (ie. integral) over the same set. See here for a recent paper on this subject.
Another surprising analogy between these two semirings arises we we try to transfer the concept of the Fourier transform to (R,+,*) to (R',min,+). It's not obvious how to interpret the exponential function in (R',min,+) but it turns out that the natural choice is to consider the ordinary linear functions (in the conventional sense) to be the correct analogue. If we then replace the exponentials with linear functions in the definition of the Fourier transform and replace the integral with an infinite min what we end up with is another familiar transform: the Legendre transform. So the Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings.
The analogy carries quite far. In classical mechanics the Legendre transform converts between the Lagrangian and Hamiltonian formulations of classical mechanics. So it changes equations of motion written in terms of (generalised) position into equations written in terms of momentum and vice versa. In quantum mechanics the Fourier transform does much the same thing: the Fourier transform of a wavefunction in space gives the wavefunction in momentum space. Sean Walston has a paper on this. I'm not sure he was aware of the semiring connection when he wrote that.
In summary we have:
| (R',min,+) | (R,+,*) |
| min(a,infinity) = a | a+0 = a |
| a+0 = a | a*1 = a |
| a+infinity = infinity | a*0 = 0 |
| Classical | Quantum |
| integral | minimisation |
| x -> k*x | x -> exp(k*x) |
| Legendre Transform | Fourier Transform |
| Hamiltonian principle of least action | Feynman path integral |
I never did understand the Legendre transform. It always seemed like this strange thing plucked out of nowhere. So it's amazing to see that in some sense it is the 'right' thing to study and is as natural as the Fourier transform. Fascinating stuff!
Labels: mathematics, physics
posted by sigfpe at Friday, October 07, 2005
13 Comments:
Fascinating stuff! But I think you have a few typos. In the text you say that "the positive infinity acts as the identity for +", but I think you mean for min. You have this stated correctly in the table at the bottom, but it seems that the first three rows of the table have the left and right columns switched.
I don't know too much about the physics stuff you talk about here, but it's nice that the concepts from analysis can be generalized to apply to the other semiring!
Oops! Mistake corrected. Thanks.
I think I remember you're a probability guy. This paper approaches the subject from the angle of probability theory, albeit in French. Confusingly they call the Legendre transform the Fenchel transform. You can extend the table to have the rows:
Quadratic form <--> Normal distribution
inf-convolution <--> Convolution
Or even this paper which explicitly talks about a 'duality'.
I am in fact interested in probability, but I've been coming more at the foundations, from the philosophical angle, and still need some more work on measure theory and mathematical probability before I'll understand what's going on in those papers, but thanks! (Set theory is probably the area of mathematics proper that I'm most competent in.)
I'm wondering if there is a connection to image recognition here. Image can be recognized by it's fourier transform or by feature points. I think feature points have some realtion to Legandre transform
There is almost certainly some application of some of this stuff to feature recognition. Both convolution (eg. blurring and sharpening filters) and inf-convolution (eg. in so called 'morphological' filters) play an important role in feature recognition algorithms.
Just minor remark: the Fenchel transform is in fact a better name for what you quote. Legendre was working in the golden age when all functions were (infinitely) differentiable, and his original definition involved inverting the gradient of a (smooth) convex function rather than taking maxima. The modern definition allows for nonsmooth functions and was first given in the 1D by the French mathematician Mandelbrojt (an uncle of Benoit Mandelbrot) in the 1930s. It was extended to multivariate case by Fenchel in the 1940s.
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what you're referring to is the area of "tropical mathematics" or "idempotent mathematics". David Corfield and John Baez have a nice discussion of this.
Are "integral" and "minimisation" swapped in the table?
Also, I think it is strange to say that integration is the "quantum" version of minimization, aren't we forgetting about classical statistical mechanics? See the previously linked section by John Baez:
http://math.ucr.edu/home/baez/qg-spring2004/discussion.html#idempotent
Does the Laplace transform deserve a mention here?
Also, the link to Sean Walston's paper is broken - do you remember the title? Thanks
I recently tried to find the Sean Walston paper again but was unable to track it down. Sorry, I don't know the title.
Went to a great lecture about Tropical Math by Berndt Sturmfels, visiting Caltech from Berkeley (he was). I intend to do Tropical Math for my Latino Algebra 1 students this afternoon. Get them to think outside the box...
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