QM without complex numbers

The nature of complex numbers in QM turned up in a recent discussion, and I got called a stupid hack for questioning their relevance. Mainly for therapeutic reasons, I wrote up my take on the issue:

On the Role of Complex Numbers in Quantum Mechanics

Motivation

It has been claimed that one of the defining characteristics that separate the quantum world from the classical one is the use of complex numbers. It's dogma, and there's some truth to it, but it's not the whole story:

While complex numbers necessarily turn up as first-class citizen of the quantum world, I'll argue that our old friend the reals shouldn't be underestimated.

A bird's eye view of quantum mechanics

In the algebraic formulation, we have a set of observables of a quantum system that comes with the structure of a real vector space. The states of our system can be realized as normalized positive (thus necessarily real) linear functionals on that space.

In the wave-function formulation, the Schrödinger equation is manifestly complex and acts on complex-valued functions. However, it is written in terms of ordinary partial derivatives of real variables and separates into two coupled real equations - the continuity equation for the probability amplitude and a Hamilton-Jacobi-type equation for the phase angle.

The manifestly real model of 2-state quantum systems is well known.

Complex and Real Algebraic Formulation

Let's take a look at how we end up with complex numbers in the algebraic formulation:

We complexify the space of observables and make it into a $C^*$-algebra. We then go ahead and represent it by linear operators on a complex Hilbert space (GNS construction).

Pure states end up as complex rays, mixed ones as density operators.

However, that's not the only way to do it:

We can let the real space be real and endow it with the structure of a Lie-Jordan-Algebra. We then go ahead and represent it by linear operators on a real Hilbert space (Hilbert-Schmidt construction).

Both pure and mixed states will end up as real rays. While the pure ones are necessarily unique, the mixed ones in general are not.

The Reason for Complexity

Even in manifestly real formulations, the complex structure is still there, but in disguise:

There's a 2-out-of-3 property connecting the unitary group $U(n)$ with the orthogonal group $O(2n)$, the symplectic group $Sp(2n,\mathbb R)$ and the complex general linear group $GL(n,\mathbb C)$: If two of the last three are present and compatible, you'll get the third one for free.

An example for this is the Lie-bracket and Jordan product: Together with a compatibility condition, these are enough to reconstruct the associative product of the $C^*$-algebra.

Another instance of this is the Kähler structure of the projective complex Hilbert space taken as a real manifold, which is what you end up with when you remove the gauge freedom from your representation of pure states:

It comes with a symplectic product which specifies the dynamics via Hamiltonian vector fields, and a Riemannian metric that gives you probabilities. Make them compatible and you'll get an implicitly-defined almost-complex structure.

Quantum mechanics is unitary, with the symplectic structure being responsible for the dynamics, the orthogonal structure being responsible for probabilities and the complex structure connecting these two. It can be realized on both real and complex spaces in reasonably natural ways, but all structure is necessarily present, even if not manifestly so.

Conclusion

Is the preference for complex spaces just a historical accident? Not really. The complex formulation is a simplification as structure gets pushed down into the scalars of our theory, and there's a certain elegance to unifying two real structures into a single complex one.

On the other hand, one could argue that it doesn't make sense to mix structures responsible for distinct features of our theory (dynamics and probabilities), or that introducing un-observables to our algebra is a design smell as preferably we should only use interior operations.

While we'll probably keep doing quantum mechanics in terms of complex realizations, one should keep in mind that the theory can be made manifestly real. This fact shouldn't really surprise anyone who has taken the bird's eye view instead of just looking throught the blinders of specific formalisms.

The complex numbers in quantum mechanics are mostly a fake. They can be replaced everywhere by real numbers, but you need to have two wavefunctions to encode the real and imaginary parts. The reason is just because the eigenvalues of the time evolution operator $e^{iHt}$ are complex, so the real and imaginary parts are degenerage pairs which mix by rotation, and you can relabel them using i.

The reason you know i is fake is that not every physical symmetry respects the complex structure. Time reversal changes the sign of "i". The operation of time reversal does this because it is reversing the sense in which the real and imaginary parts of the eigenvectors rotate into each other, but without reversing the sign of energy (since a time reversed state has the same energy, not negative of the energy).

This property means that the "i" you see in quantum mechanics can be thought of as shorthand for the matrix (0,1;-1,0), which is algebraically equivalent, and then you can use real and imaginary part wavefunctions. Then time reversal is simple to understand--- it's an orthogonal transformation that takes i to -i, so it doesn't commute with i.

The proper way to ask "why i" is to ask why the i operator, considered as a matrix, commutes with all physical observables. In other words, why are states doubled in quantum mechanics in indistinguishable pairs. The reason we can use it as a c-number imaginary unit is because it has this property. By construction, i commutes with H, but the question is why it must commute with everything else.

One way to understand this is to consider two finite dimensional systems with isolated Hamiltonians $H_1$ and $H_2$, with an interaction Hamiltonian $f(t)H_i$. These must interact in such a way that if you freeze the interaction at any one time, so that $f(t)$ rises to a constant and stays there, the result is going to be a meaningful quantum system, with nonzero energy. If there is any point where $H_i(t)$ doesn't commute with the i operator, there will be energy states which cannot rotate in time, because they have no partner of the same energy to rotate into. Such states must be necessarily of zero energy. The only zero energy state is the vacuum, so this is not possible.

You conclude that any mixing through an interaction hamiltonian between two quantum systems must respect the i structure, so entangling two systems to do a measurement on one will equally entangle with the two state which together make the complex state.

It is possible to truncate quantum mechanics (at least for sure in a pure bosnic theory with a real Hamiltonian, that is, PT symmetric) so that the ground state (and only the ground state) has exactly zero energy, and doesn't have a partner. For a bosonic system, the ground state wavefunction is real and positive, and if it has energy zero, it will never need the imaginary partner to mix with. Such a truncation happens naturally in the analytic continuation of SUSY QM systems with unbroken SUSY.

If you don't like complex numbers, you can use pairs of real numbers $(x,y)$. You can "add" two pairs by $(x,y)+(z,w) = (x+z,y+w)$, and you can "multiply" two pairs by $(x,y) * (z,w) = (xz-yw, xw+yz)$. (If don't think that multiplication should work that way, you can call this operation "shmultiplication" instead.)

Now you can do anything in quantum mechanics. Wavefunctions are represented by vectors where each entry is a pair of real numbers. (Or you can say that wavefunctions are represented by a pair of real vectors.) Operators are represented by matrices where each entry is a pair of real numbers, or alternatively operators are represented by a pair of real matrices. Shmultiplication is used in many formulas. Etc. Etc.

I'm sure you see that these are exactly the same as complex numbers. (see Lubos's comment: "a contrived machinery that imitates complex numbers") They are "complex numbers for people who have philosophical problems with complex numbers". But it would make more sense to get over those philosophical problems. :-)