## Real and quaternionic quantum mechanics

Wrapping up the discussion about quaternionic quantum mechanics, in this theory the inner product $$<f|g> = \int d^3 x \bar{f}(x,t) g(x,t)$$ decomposes into a complex inner product and a symplectic inner product:

$$<f|g> = {<f|g>}_{C} + {<f|g>}_{S}$$

Here "f bar" means complex conjugation with respect to all imaginary elements: i, j, k.

If we consider quaternionic wavefunction $$f$$ decomposition: $$f = f_{\alpha} + j f_{\beta}$$ where $$f_{\alpha} = f_0 + i f_1$$ and $$f_{\beta} = f_2 - i f_3$$ then:

$${<f|g>}_{C} = \int d^3 x {f}^{*}_{\alpha}(x,t) {g}_{\alpha}(x,t) + {f}^{*}_{\beta}(x,t) {g}_{\beta}(x,t)$$

and

$${<f|g>}_{S} = \int d^3 x {f}_{\alpha}(x,t) {g}_{\beta}(x,t) - {f}_{\beta}(x,t) {g}_{\alpha}(x,t)$$

Therefore in quaternionic quantum mechanics the probabilities are different then in complex quantum mechanics because it includes the symplectic part:

$${|<f|g>|}^2\ = {|{<f|g>}_{C}|}^2 + {|{<f|g>}_{S}|}^2$$

Now real quantum mechanics is defined over the real numbers and we can compare it with complex quantum mechanics. If we consider the complex quantum mechanics decomposition of the wavefunction:

$$f = f_0+ i f_1$$

the complex inner product is:

$$<f|g> = {<f|g>}_{C} + {<f|g>}_{I}$$

with

$${<f|g>}_{R} = \int d^3 x {f}_{0}(x,t) {g}_{0}(x,t) + {f}_{1}(x,t) {g}_{1}(x,t)$$
and
$${<f|g>}_{I} = \int d^3 x {f}_{0}(x,t) {g}_{1}(x,t) - {f}_{1}(x,t) {g}_{0}(x,t)$$

Then

$${|<f|g>|}^2\ = {|{<f|g>}_{R}|}^2 + {|{<f|g>}_{I}|}^2$$

and real quantum mechanics has only this term: $${|{<f|g>}_{R}|}^2$$

Now for the main problem of real quantum mechanics: in real quantum mechanics there are no energy eigenstates. Because of this we need to embed real quantum mechanics in complex quantum mechanics!!!

Quaternionic quantum mechanics does not have such problems, but in quaternionic quantum mechanics a De Finetti theorem does not hold. The root cause of it is the fact that quaternionic quantum mechanics does not respect the tensor product.

In conclusion, both real and quaternionic quantum mechanics have very limited usefulness in describing nature and the most general way to express quantum mechanics is over complex numbers.

Now the million dollar question: are the quantum mechanics number systems exhausted by the reals, complex numbers, or quaternions? The answer is no and there is at least one more: a direct sum of two SL(2,C). This will lead to Dirac's equation of the electron. Please stay tuned.

## Is Quaternionic Quantum Mechanics Detectable?

Let us start by explaining why quaternionic quantum mechanics may be interesting for describing nature despite its problem with the tensor product.

There are several arguments put forward, but I think only two of them have merit. The first obvious usage can be in justifying the $$SU(2)$$ gauge group of the Standard Model which can be achieved in a similar fashion of how $$U(1)$$ (the gauge group of electromagnetism) arises when one considers local symmetry.

The second potential usage stems from the peculiar behavior of quaternionic quantum mechanics: the ground energy level is uniquely determined and the zero point energy cannot be freely shifted like in complex quantum mechanics. The obvious application of this is in solving the puzzle of the cosmological constant which by naive field theory arguments should be $${10}^{120}$$ times larger than the observed value.

A pure state in quaternionic quantum mechanics is defined only up to a quaternionic phase (a unit quaternion) and one may ask if such phases which are different than complex wavefunction phases are experimentally detectable. In fact Asher Peres

 Asher Peres

proposed such an experiment: pass a neutron beam through slabs of dissimilar materials and search for the non-commutativity of the phase shift when the slabs are reversed. The experiment was performed and the answer was negative. The theoretical clarification came later on from Adler's analysis which showed that the S-matrix quaternionic scattering is in fact indistinguishable from the usual complex quantum mechanics scattering.

This result is not surprising given the previous post: quaternionic quantum mechanics can be understood as a constrained complex quantum mechanics and there the square roof of negative one is represents a map between observables (hermitean operators) and generators of continuous symmetries (anti-hermitean operators). This map is also known as "dynamic correspondence".

Before comparing quaternionic quantum mechanics with real and complex quantum mechanics there is one last result of relative importance. Wigner's theorem states that in complex quantum mechanics symmetries can be represented by a unitary or anti-unitary transformation. Emch, Piron, Uhlhorn and Bargman generalized this to the quaternionic quantum mechanics case and here there are only quaternionic-unitary transformations.

## (part 2)

Continuing the quaternionic discussion, let us see when one might encounter it. Here I will follow the discussion in "Geometry of Quantum States" by Ingemar Bengtsson and Karol Zyczkowski. The idea is that of a time reversal.

We start in the usual complex quantum mechanics and here by Wigner's theorem we have that every symmetry is represented by a unitary or anti-unitary transformation. So suppose we have a time reversal operator $$\Theta$$. When we reverse time, from the time evolution equation in the Heisenberg picture this is equivalent with $$i$$ changing signs, and therefore $$\Theta$$ must be an anti-unitary transformation.

If the system is invariant under time reversal, we have:

$$\langle \Psi| \Phi \rangle = \langle \Theta\Psi| \Theta\Phi \rangle$$

and this means that there are two options for $$\Theta$$: $$\Theta^2 = \pm 1$$.

Now the discussion depends on the angular momentum of the system. For fermions $$\Theta^2 = - 1$$. If we cannot tell the direction of time by any measurement, the observables commute with $$\Theta$$:

$$[O, \Theta] = 0$$

and this defines a superselection rule. Then one can define:

$$i, \Theta, i\Theta\ = i, j, k$$

the quaternionic imaginary elements and one arrives at the quaternionic projective space. Here is how quaternionic quantum mechanics can arise.

However, because in this case one talks about superselection rules, composing two quaternionic systems breaks the superselection constraint and there are problems defining the tensor product. We'll talk about this next time.

## (part 1)

I will start a new series about the number system of quantum mechanics. Quantum mechanics can be expressed over real numbers, complex numbers, quaternions, and SL(2,C). I will simply follow the literature and try to present the interesting results which will help better understand the usual complex number formalism.

The standard reference for quaternionic quantum mechanics is Adler's monograph: Quaternionic Quantum Mechanics and Quantum Fields

First, what is a quaternion? It is one of the 4 normed division number systems which consists of the elements of the form:

$$w = a+ ix + jy + kz$$ with $$a,x,y,z \in R$$
where $$i^2 = j^2 = k^2 = -1$$ and $$ij=k, jk=i, ki=j$$

 Quaternionic multiplication

Just like in complex quantum mechanics, a physical state is defined only up to a phase which here is a unit quaternion:

$$| \psi \rangle = \{|\psi \omega \rangle : |\omega| = 1\}$$

This works because the probability of the quantum transition between states $$\psi, \phi$$ is given by the usual rule:

$$P = {|\langle \psi | \phi \rangle |}^2$$

Since unlike complex numbers quaternions are non-commutative ($$ij = -ji \ne ji$$) we have to be careful on the position of the numbers in the ket-bra notation. By convention we say:

$$|\psi \omega \rangle = |\psi \rangle \omega$$

and we will have the following linearity condition for an operator:

$$O (|\psi \rangle \omega) = (O|\psi\rangle) \omega$$

If $$1$$ is the identity operator, let us define:

$$1 = E_0$$
$$I = E_1 = i1$$
$$J=E_2 = j1$$
$$K=E_3 = k1$$

and an operator $$O$$ has the decomposition: $$O = O_0 + I O_1 + J O_2 + K O_3$$ where:

$$O_0 = 1/4 (O - IOI -JOJ -KOK)$$
$$O_1 = 1/4 (IO + OI -JOK + KOJ)$$
$$O_2 = 1/4 (JO + OJ -KOI + IOK)$$
$$O_3 = 1/4 (KO + OK -IOJ + JOI)$$

Similarly a quaternionic wavefunction can be decomposed as follows:

$$|\psi \rangle = |\psi_0 \rangle + I |\psi_1 \rangle + J |\psi_2 \rangle + K |\psi_3 \rangle$$

where:

$$|\psi_0 \rangle = 1/4(|\psi\rangle - I |\psi\rangle i - J|\psi\rangle j -K|\psi\rangle k)$$
$$|\psi_1 \rangle = -1/4(I |\psi\rangle + |\psi\rangle i - J|\psi\rangle k + K|\psi\rangle j)$$
$$|\psi_2 \rangle = -1/4(J|\psi\rangle + |\psi\rangle j - K|\psi\rangle i + I|\psi\rangle k)$$
$$|\psi_3 \rangle = -1/4(K|\psi\rangle + |\psi\rangle k - I|\psi\rangle j + J|\psi\rangle i)$$

To be continued ...