MathPhys

In the standard Hermitian quantum mechanics, the space of states of a quantum mechanical system forms a complex Hilbert space in general. But often the space of states becomes a real vector space. When the space of states is a real vector space, an alternative quantum mechanics may be considered.

Let $\mathbb{V}$ be the complex vector space of states of a quantum mechanical system. Define a map $J: \mathbb{V}\longrightarrow \mathbb{V}$ by
$$J\psi=\bar\psi$$
for every $\psi\in \mathbb{V}$. Then $J$ is an anti-linear involution called a real structure on $\mathbb{V}$. A real structure on a complex vector space is a way to decompose the complex vector space $\mathbb{V}$ into the direct sum of two real vector spaces. Any $\psi\in \mathbb{V}$ may be written as
$$\psi=\psi^++\psi^-,$$
\begin{align*}
\psi^+&:=\frac{1}{2}(\psi+J\psi)=\frac{1}{2}(\psi+\bar\psi)=\psi_{\mathrm{re}},\\
\psi^-&:=\frac{1}{2}(\psi-J\psi)=\frac{1}{2}(\psi-\bar\psi)=i\psi_{\mathrm{im}}.
\end{align*}
$J$ satisfies the properties:
\begin{align*}
J\psi^+&=\psi^+,\\
J\psi^-&=-\psi^-.
\end{align*}
Hence, one obtains a direct sum of vector spaces
$$\mathbb{V}=\mathbb{V}^+\oplus \mathbb{V}^-,$$
where
\begin{align*}
\mathbb{V}^+&=\{\psi\in \mathbb{V}: J\psi=\psi\},\\
\mathbb{V}^-&=\{\psi\in \mathbb{V}: J\psi=-\psi\}.
\end{align*}
Both $\mathbb{V}^+$ and $\mathbb{V}^-$ are real vector spaces. $\mathbb{V}^+$ is isomorphic to $\mathbb{V}^-$ via the isomorphism $\psi\longmapsto i\psi$. So,
$$\dim_{\mathbb{R}}\mathbb{V}^+=\dim_{\mathbb{R}}\mathbb{V}^-=\dim_{\mathbb{C}}\mathbb{V}$$
if $\dim_{\mathbb{C}}\mathbb{V}<\infty$. Let $\mathbb{V}^+=\mathbb{V}_{\mathbb{R}}$. Then $\mathbb{V}^-$ can be written as $i\mathbb{V}_{\mathbb{R}}$, so $\mathbb{V}$ may be viewed as the complexification of $\mathbb{V}_{\mathbb{R}}$
$$\mathbb{V}_{\mathbb{R}}^\mathbb{C}=\mathbb{V}_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}.$$
The real structure $J$ may be used to define an inner product $\langle\ ,\ \rangle$ on $\mathbb{V}$: For any $\varphi,\psi\in \mathbb{V}$,
\begin{align*}
\langle\varphi,\psi\rangle&=\int (J\varphi)\psi dV\\
&=\int\bar\varphi\psi dV.
\end{align*}
This inner product is of course the standard Hermitian product known as Dirac braket $\langle\ |\ \rangle$ to physicists. Therefore, the real structure gives rise to the Hilbert space structure on $\mathbb{V}$ as in the standard Hermitian quantum mechanics.

In studying a quantum mechanical system, it is not necessary to use complex-valued state functions but one may instead use real-valued state functions. For example, let us consider the Hamiltonian for quantum harmonic oscillator
$$\hat H=-\frac{\hbar^2}{2m}\frac{\partial}{\partial x^2}+\frac{m}{2}\omega^2\hat x^2.$$
The stationary Schrödinger equation $\hat H\psi=E\psi$ determines the eigenstates
$$\psi_n(\xi)=2^{-\frac{n}{2}}\pi^{-\frac{1}{4}}(n!)^{-\frac{1}{2}}e^{-\frac{\xi^2}{2}}H_n(\xi),$$
where $\xi=\sqrt{\frac{m\omega}{\hbar}}x$, and eigenvalues (energies)
$$E_n=\left(n+\frac{1}{2}\right)\hbar\omega$$
for $n=0,1,2,\cdots$. $H_n(x)$ are the Hermite polynomials and they satisfy the parity relation
$$H_n(x)=(-1)^nH_n(-x).$$
The eigenstates $\psi_n(\xi)$ forms a real Hilbert space with the usual Hermitian product (Dirac braket).

This time, let $\mathbb{V}$ be the real vector space of states of a quantum mechanical system. Let $\varphi:\mathbb{R}^3\longrightarrow\mathbb{R}$. $\varphi$ is said to be even under parity if $\varphi(-\mathbf{r})=\varphi(\mathbf{r})$. $\varphi$ is said to be odd under parity if $\varphi(-\mathbf{r})=-\varphi(\mathbf{r})$. Any real-valued function defined on $\mathbb{R}^3$ may be written as the sum of an even function and an odd function. Any $\varphi(\mathbf{r})\in \mathbb{V}$ can be written as
$$\varphi(\mathbf{r})=\frac{1}{2}[\varphi(\mathbf{r})+\varphi(-\mathbf{r})]+\frac{1}{2}[\varphi(\mathbf{r})-\varphi(-\mathbf{r})].$$
$\frac{1}{2}[\varphi(\mathbf{r})+\varphi(-\mathbf{r})]$ is an even function and $\frac{1}{2}[\varphi(\mathbf{r})-\varphi(-\mathbf{r})]$ is an odd function. Define a map $J: \mathbb{V}\longrightarrow \mathbb{V}$ by
$$J\varphi(\mathbf{r})=\varphi(-\mathbf{r})$$
for every $\varphi(\mathbf{r})\in \mathbb{V}$ i.e. $J$ is the parity. Then $J$ is a linear involution. Any $\varphi(\mathbf{r})\in \mathbb{V}$ may be written as
$$\varphi(\mathbf{r})=\varphi^+(\mathbf{r})+\varphi^-(\mathbf{r}),$$
where
\begin{align*}
\varphi^+(\mathbf{r})&=\frac{1}{2}[\varphi(\mathbf{r})+J\varphi(\mathbf{r})]=\frac{1}{2}[\varphi(\mathbf{r})+\varphi(-\mathbf{r})],\\
\varphi^-(\mathbf{r})&=\frac{1}{2}[\varphi(\mathbf{r})-J\varphi(\mathbf{r})]=\frac{1}{2}[\varphi(\mathbf{r})-\varphi(-\mathbf{r})].
\end{align*}
We have
\begin{align*}
J\varphi^+(\mathbf{r})&=\varphi^+(\mathbf{r}),\\
J\varphi^-(\mathbf{r})&=-\varphi^-(\mathbf{r}).
\end{align*}
The map $J$ is called the fundamental symmetry (see [1] for example). One gets a direct sum of vector spaces
$$\mathbb{V}=\mathbb{V}^+\oplus \mathbb{V}^-,$$
where
\begin{align*}
\mathbb{V}^+&=\{\varphi\in \mathbb{V}: J\varphi=\varphi\},\\
\mathbb{V}^-&=\{\varphi\in \mathbb{V}: J\varphi=-\varphi\}.
\end{align*}
If $\varphi(\mathbf{r})$ is even, then $\nabla\cdot\varphi(\mathbf{r})$ is odd and if $\varphi(\mathbf{r})$ is odd, then $\nabla\cdot\varphi(\mathbf{r})$ is even. If we say two functions $\varphi_1(\mathbf{r})$ and $\varphi_2(\mathbf{r})$ are identical if $\varphi_1-\varphi_2$ is constant, then $\mathbb{V}^+$ is isomorphic to $\mathbb{V}^-$ via the linear map $\nabla\cdot\varphi(\mathbf{r})$.

The fundamental symmetry may be used to define an inner product $\langle\ ,\ \rangle$ on $\mathbb{V}$ as follows: For any $\varphi(\mathbf{r}),\psi(\mathbf{r})\in \mathbb{V}$,
\begin{align*}
\langle\varphi(\mathbf{r}),\psi(\mathbf{r})\rangle&=\int (J\varphi(\mathbf{r}))\psi(\mathbf{r})dV\\
&=\int\varphi(-\mathbf{r})\psi(\mathbf{r})dV.
\end{align*}
For any $\varphi^+(\mathbf{r})\in \mathbb{V}^+$, $\varphi^-(\mathbf{r})\in \mathbb{V}^-$, we have
\begin{align*}
\langle\varphi^+(\mathbf{r}),\varphi^+(\mathbf{r}\rangle&=\int[\varphi^+(\mathbf{r})]^2dV\geq 0,\\
\langle\varphi^-(\mathbf{r}),\varphi^-(\mathbf{r})\rangle&=-\int[\varphi^-(\mathbf{r})]^2dV\leq 0
\end{align*}
We may assume that each $\varphi(\mathbf{r})$ is separable i.e. $\varphi(\mathbf{r})=X(x)Y(y)Z(z)$. ($\varphi(\mathbf{r})$ is a solution of the stationary Schrödinger equation $\hat H\varphi=E\varphi$ and this is a condition that can be imposed when we solve the equation.) Furthermore, we may assume that each of $X(x)$, $Y(y)$ and $Z(z)$ is either even or odd. Since $\varphi(-\mathbf{r})=X(-x)Y(-y)Z(-z)$, if $\varphi(-\mathbf{r})$ is odd, then at least one of $X(x)$, $Y(y)$, $Z(z)$ must be odd. For any $\varphi^+(\mathbf{r})\in \mathbb{V}^+$, $\varphi^-(\mathbf{r})\in \mathbb{V}^-$, $\varphi^+(\mathbf{r})\varphi^-(\mathbf{r})$ is odd. Let us write $\varphi^+(\mathbf{r})\varphi^-(\mathbf{r})=X(x)Y(y)Z(z)$ and say $X(x)$ is odd. Then
\begin{align*}
\langle\varphi^+(\mathbf{r}),\varphi^-(\mathbf{r}\rangle&=\int \varphi^+(\mathbf{r})\varphi^-(\mathbf{r})dV\\
&=\int_{-\infty}^\infty X(x)dx\int_{-\infty}^\infty Y(y)dy\int_{-\infty}^\infty Z(z)dz\\
&=0.
\end{align*}
Hence, $\varphi^+(\mathbf{r})$ and $\varphi^-(\mathbf{r})$ are orthogonal for all $\varphi^+(\mathbf{r})\in \mathbb{V}^+$, $\varphi^-(\mathbf{r})\in \mathbb{V}^-$, and so the direct sum $\mathbb{V}^+\oplus \mathbb{V}^-$ can be replaced by the orthogonal sum $\mathbb{V}^+\dotplus \mathbb{V}^-$. Note that $\langle\ ,\ \rangle$ is indefinte, so the fundamental symmetry gives rise to a Krein space structure rather than a Hilbert space structure on $\mathbb{V}$. Here, we are using a particular fundamental symmetry which is the parity, however our discussion may be generalised in terms of an arbitrary fundamental symmetry $J$. Although it may seem redundant, physically the inner product needs to be redefined as
\begin{align*}
\langle\varphi(\mathbf{r}),\psi(\mathbf{r})\rangle&=\int\overline{J\varphi(\mathbf{r})}\psi(\mathbf{r})dV\\
&=\int\bar\varphi(\mathbf{r})\psi(\mathbf{r})dV
\end{align*}
for any $\varphi(\mathbf{r}),\psi(\mathbf{r})\in \mathbb{V}$. The reason is that in quantum mechanics $\langle\ ,\ \rangle$ often interacts with operators that may be complex.

In the above example of quantum harmonic oscillator, let $\psi_n^+(\xi)=\psi_{2n}(\xi)$ and $\psi^-_{2n+1}(\xi)$ for $n=0,1,2,\cdots$. Then we have $\langle\psi^+_m,\psi^+_n\rangle=\delta_{mn}$, $\langle\psi^-_m,\psi^-_n\rangle=-\delta_{mn}$, and $\langle\psi^+_m,\psi^-_n\rangle=0$. Let $\mathbb{V}^+$ and $\mathbb{V}^-$ be spaces spanned by the orthogonal bases $\mathcal{B}^+=\{\psi^+_n(\xi): n=0,1,2,\cdots\}$ and $\mathcal{B}^-=\{\psi^-_n(\xi): n=0,1,2,\cdots\}$, respectively. Then $\mathbb{V}=\mathbb{V}^+\dotplus \mathbb{V}^-$ is a Krein space.

When the state functions of a quantum mechanical system are real-valued, the state functions may form a real Hilbert space or a Krein space depending on the choice between the inner products we discussed above. Hence, one may expect to have a quantum theory alternative, in which case a Krein space arises, to the standard Hermitian quantum mechanics. This alternative quantum theory is PT-Symmetric Quantum Mechanics which was first introduced by Carl M. Bender. PT-symmetric quantum mechanics has been promoted as a new quantum theory which admits a certain type of complex non-Hermitian Hamiltonians, the so-called PT-symmetric Hamiltonians. I recently showed that PT-symmetric quantum mechanics is indeed a Hermitian quantum mechanics and that those complex non-Hermitian (they are actually Hermitian) Hamiltonians are not really physical. (They would result the violation of unitarity.) After all, once disregarding those unphysical complex Hamiltonians, this so-called PT-symmetric quantum mechanics is not that much different at all from the standard Hermitian quantum mechanics. I will write more about this some other time. In the meantime, interesting readers may be referred to my recent papers [2], [3] on the issue.

Update 1: To clarify, here I meant a PT-symmetric Hamiltonian being Hermitian by it being self-adjoint with respect to the indefinite Hermitian inner product above. That’s why I said “PT-symmetric quantum mechanics is a Hermitian quantum mechanics.”

Update 2: My stance on PT-symmetric quantum mechanics has changed as I have a much better understanding of what’s going on. I considered complex PT-symmetric Hamiltonians were unphysical due to physicists’ insistance on using definite inner product because they want to interpret $|\psi|^2$ as probability like the case in ordinary quantum mechanics. But if we can relax about the meaning of $|\psi|^2$ as not necessarily probability but something that needs to be preserved under time evolution, the issue of unitarity violation can be resolved and complex PT-symmetric Hamiltonians become physical. It is possible that the interpretation of $|\psi|^2$ being probability is valid only in the ordinary quantum mechanics and if there is a new formulation of quantum mechanics, perhaps it may require a new set of rules and interpretations.

References:

[1] János Bognár, Indefinite Inner Product Spaces, Springer-Verlag 1974
[2] Sungwook Lee, $PT$-symmetric quantum mechanics is a Hermitian quantum mechanics, arXiv:1312.7738 [quant-ph]
[3] Sungwook Lee, On Finite $J$-Hermitian Quantum Mechanics, arXiv:1401.5149 [quant-ph]

Theory of Photons (Electromagnetism) and Wave-Particle Duality of Light

Earlier investigations on the nature of light show that, light must be described by electromagnetic waves or by particles (wave-particle duality). de Broglie hypothesized that what is true for photons should be valid for any particle. We may assign a particle with mass $m$, propagating uniformly with velocity $v$ through field-free space, an energy $E$ and momentum $\mathbf{p}$. In the wave picture, the same particle may be described by a frequency $\omega$ and a wave vector $\mathbf{k}$.  We require that these quantities satisfy the equations
$$E=\hbar\omega,\ \mathbf{p}=\hbar\mathbf{k}.$$
In fact, these equations are satisfied by the photon and that a photon is described by the plane wave
$$\psi(\mathbf{r},t)=A\exp[i(\mathbf{k}\cdot\mathbf{r}-\omega t)].$$
Following de Broglie, to every free particle, a plane wave shown above is assigned. So, naturally quantum mechanics was formulated in terms of complex numbers.

The Trouble with Quantum Mechanics

Quantum mechanics as it was formulated appears to be seamless and it has had decades of success with its great advancement as theory of matter and also with its so many wonderful applications in science and engineering. Nevertheless, there are still some fundamental issues with current quantum mechanics. One issue is about antiparticles. In non-relativistic quantum mechanics, there is no clear indication that antiparticles exist. In relativistic quantum mechanics, antiparticles are considered as ones that have negative energies (due to P.A.M. Dirac) or equivalently ones that travel back in time. (Since wave functions themselves are not observables, those particles travel back in time do not violate causality.) But then this may imply that antiparticles have negative rest masses. So to me, considering antiparticles as particles that have negative energies appears to be unnatural and unphysical. Another issue is about the path integral. In quantum mechanics, the amplitude of a particle to propagate from a point $q_I$ to a point $q_F$ in time $T$ is given by
\begin{equation}\label{eq:pathint}\langle q_F|e^{-\frac{i}{\hbar}\hat HT}|q_I\rangle=\int Dq(t)e^{\frac{i}{\hbar}\int_0^TdtL(\dot q,q)}\end{equation}
where $L(\dot q,q)$ is the Lagrangian
$$L(\dot q,q)=\frac{m}{2}\dot q^2-V(q)$$
and $Dq(t)$ is the Feynman measure given by
$$\int Dq(t):=\lim_{N\to\infty}\left(\frac{-im\hbar}{2\pi\delta t}\right)^{\frac{N}{2}}\left(\prod_{k=1}^{N-1}\int dq_k\right)$$
with $\delta t=\frac{T}{N}$. This path integral, while it makes perfect sense physically, does not converge due to the oscillatory factor appeared as the integrand. What Physicists do about this problem is to take the Wick rotation $t\mapsto it$ which turns Minkowski spacetime to Euclidean spacetime. Accordingly, the path integral turns into Euclidean path integral
$$\langle q_F|e^{-\frac{i}{\hbar}\hat HT}|q_I\rangle=\int Dq(t)e^{-\frac{1}{\hbar}\int_0^TdtL(\dot q,q)}.$$
The integrand becomes a decaying exponential whose maximum value occurs at the minimum of the Euclidean action. Most physicists appear to be satisfied with this resolution, however to me it is troublesome that the path integral \eqref{eq:pathint} cannot be calculated in actual spacetime and that it must be calculated in Euclidean spacetime which is not physical spacetime. Besides, most Euclidean solutions are approximations and there is no guarantee that these solutions will be stable when they are brought to Minkowski spacetime. Furthermore, analytic continuation via Wick rotation works when the spacetime is flat. So Euclideanization will have a problem when the spacetime is curved i.e. gravitation is considered.

$i=\sqrt{-1}$ is the Problem!

I believe that this issue with the path integral has its origin in the way our current quantum mechanics was built. It was built upon complex numbers! And the reason complex numbers entered in the formulation of quantum mechanics is that the founders used the theory of photons (electromagnetism) to build the theory of matter, i.e. theory that explains the physics of other particles, in particular of massive particles such as electrons. However, it may be that complex numbers are really for light (photons) and they are not meant for describing other particles, especially massive particles.

Complex Numbers are Really for Light

Maxwell’s equations in vacuum are:
\begin{align*}
\nabla\cdot\mathbf{B}&=0,\ \nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}=0\\
\nabla\cdot\mathbf{E}&=0,\ \nabla\times\mathbf{B}-\frac{\partial\mathbf{E}}{\partial t}=0
\end{align*}
The transformation
$$\mathbf{B}\mapsto\mathbf{E},\ \mathbf{E}\mapsto -\mathbf{B}$$
takes the first pair of equations to the second and vice versa. This symmetry is called Electric-Magnetic Duality. The duality hints that the electric and magnetic fields are part of a unified whole, the electromagnetic field. Let us introduce a complex-valued vector field
$$\mathcal{E}=\mathbf{E}+i\mathbf{B}.$$
Then the duality amounts to the transformation
$$\mathcal{E}\mapsto -i\mathcal{E}$$
and the vacuum Maxwell’s equations boil down to two equations for $\mathcal{E}$:
$$\nabla\cdot\mathcal{E}=0,\ \nabla\times\mathcal{E}=i\frac{\partial\mathcal{E}}{\partial t}.$$
Let $\mathbf{k}$ be a vector in $\mathbb{R}^3$ and let $\omega=|\mathbf{k}|$. Fix $\mathbf{E}\in\mathbb{C}^3$ with $\mathbf{E}\cdot\mathbf{k}=0$ and $\mathbf{E}\times\mathbf{k}=i\omega\mathbf{E}$. Then the plane wave
$$\mathcal{E}(\mathbf{r},t)=\mathbf{E}\exp[i(\mathbf{k}\cdot\mathbf{r}-\omega t)]$$
satisfies the vacuum Maxwell’s equations. That is, complex-valued plane wave functions may describe light (electromagnetic waves).

Here is another example that shows complex numbers are for light. The light cone in Minkowski spacetime $\mathbb{R}^{3+1}$ is the hyperquadric
$$\mathbb{N}^3=\{(t,x,y,z)\in\mathbb{R}^{3+1}: t^2-x^2-y^2-z^2=0\}.$$
Let $\mathbb{N}^3_+$ and $\mathbb{N}^3_-$ denotes the future and the past light cones, respectively. The multiplicative group $\mathbb{R}^+$ acts on $\mathbb{N}^3_+$ and $\mathbb{N}^3_-$ respectively by scalar multiplication. The orbit spaces $\mathbb{N}^3_+/\mathbb{R}^+$ and $\mathbb{N}^3_-/\mathbb{R}^+$ are identified with the two-sphere $S^2$. Physically this means that for an observer at the origin (the event), light rays through his eye correspond to null lines through the origin and the past null directions constitute the field of vision of the observer which is the two-sphere $S^2$. This two-sphere is called the Celestial Sphere in astronomy. In mathematics, the two-sphere $S^2$ is the extended complex plane $\mathbb{C}\cup\{\infty\}$ called the Riemann sphere.

In the beginning, God might have said:

“Let there be complex numbers!”

Wave Functions are Real?

In current quantum physics, a wave functions itself is not considered as a physical reality but rather a manifestation of something that is both particle and wave. What if we assume that wave functions are real, say they represent actual waves in spacetime? If so, there can be wave functions that are split-complex-valued. I assert that split-complex numbers might have been the right choice for the theory of matter, theory of massless particles other than photons and particularly that of massive particles. It turns out that quantum mechanics can be completely rebuilt based upon split-complex numbers. This new quantum mechanics, split-Hermitian quantum mechanics, exhibits distinct features. In split-Hermitian quantum mechanics, antiparticles arise naturally. Remarkably, the path integral may be calculated in Minkowski spacetime without turning it into Euclidean path integral via Wick rotation. Furthermore, the path integral also can be transformed to the one looks just like Euclidean path integral, but it is still defined in Minkowski spacetime. Hence, what physicists have believe to be Euclidean path integral is a form of path integral defined in actual spacetime.

Baryon Asymmetry and Twin Universes

Split-Hermitian quantum mechanics may also offer an explanation on baryon asymmetry, i.e. an explanation as to why there aren’t as many antiparticles as particles in the universe (which is a fortunate thing for us). Although it is highly speculative at this juncture, split-Hermitian quantum mechanics appears to hint that Big Bang may have created twin (not exactly identical though) universes, one with signature $(- + + +)$ made mostly of matter (our universe) and the other with signature $(+ – ++)$ made mostly of antimatter.