It is a strict requirement that physical observables should be real-valued; the position and momentum of a particle obeying the Newtonian equations of motion are both real, and any dynamical variables derived from them are also real. The wavefunction in quantum mechanics may be complex-valued, but observables obtained averaged over the square modulus of the wavefunction turn out to be real. Occasionally, one examines the nature of singularities in the complex plane in order to judge whether the system under consideration is completely integrable or not, but in almost all cases complex variables appear only implicitly in the description of physical phenomena. Indeed, there is no serious demand or even motivation to investigate dynamics whose variables are complex.
It is a strict requirement that physical observables should be real-valued; the position and momentum of a particle obeying the Newtonian equations of motion are both real, and any dynamical variables derived from them are also real. The wavefunction in quantum mechanics may be complex-valued, but observables obtained averaged over the square modulus of the wavefunction turn out to be real. Occasionally, one examines the nature of singularities in the complex plane in order to judge whether the system under consideration is completely integrable or not, but in almost all cases complex variables appear only implicitly in the description of physical phenomena. Indeed, there is no serious demand or even motivation to investigate dynamics whose variables are complex.
The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the complex-valued quantum mechanics in that the eigen-structures of real and complex quantum systems, such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant under linear complex mapping. In other words, there is indeed no distinction between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems when viewed from a complex domain. Their eigen-structures can be made coincident through linear transformation of complex coordinates.
Abstract The general representation theory of quantum mechanics requires one to represent the states of a dynamical system by functions of a set of real variables q'r, each of which has a domain consisting of either discrete points or a continuous range of points, or possibly both together. A dynamical variable is represented by a function of two such sets of real variables q'r and q"r forming a generalized “matrix". In this paper we shall show that in certain cases it is advantageous to consider some of our variables q'r as complex variables and to suppose the representatives of states and dynamical variables to depend on them in accordance with the theory of functions of a complex variable. In the usual theory the domains of the s are the eigenvalues of certain observables q"r. This significance of the q"s of course gets lost when we consider them as complex variables, but we have, however, some beautiful mathematical features appearing instead, and we gain a considerable amount of mathematical power for the working out of particular examples.
Abstract A class of Newtonian forces, determining the acceleration F (x, y) of particles in the plane, is F =(Re F(z), Im F(z)), where z is the complex variable x + iy. Curl F is non-zero, so these forces are nonconservative. These complex curl forces correspond to completely integrable Hamiltonians that are anisotropic in the momenta, separable in z and z * but not in x and y if the curl is nonzero. The Hamiltonians can be quantised, leading to unfamiliar wavefunctions, even for the (non-curl) isotropic harmonic oscillator. The formalism provides an alternative interpretation of the analytic continuation of one-dimensional real Hamiltonian particle dynamics, where trajectories are known to exhibit intricate structure (though not chaos), and is a Hermitian alternative to non-Hermitian quantisation.
Physical observables which are measured in a particular state of a dynamical system can be conveniently represented in classical mechanics by mathematical variables, on the grounds that all have definite values in any classical state. In contrast, the statistical interpretation of wave functions describing the quantum states implies that there are uncertainties in the measurement of any two observables which can be represented by canonically conjugate variables, such as position q and momentum p q . This requires any measured dynamical observable A in a quantum state to be associated with an expectation value 〈 A 〉.
We show that a backward stochastic system with an imaginary diffusion coefficient in complex space is equivalent to a real space quantum system if it obeys a Newtonian-like dynamical law and its final spatial distribution is constrained to be in real space. The ensemble averages of the position, momentum, angular momentum and energy of the stochastic processes are the quantum mechanical expectation values of the corresponding operators. The product of the mean-square deviations of the position and momentum of the stochastic process agreees with the corresponding uncertainty relation of quantum mechanics. The equation of motion is invariant under the combined transformations of changing the time t! ¡ t and taking the complex conjugate of the equation. The stochastic motion reduces to the classical motion in the limit where the de Broglie wavelengh of the particle vanishes. PACS. 03.65.Bz – Foundations, theory of measurement, miscellaneous theories. PACS. 02.50.Ga – Markov processes. PACS. 02.50.Cw – Probability theory.
It is known from quantum mechanics that particles are associated with wave functions, and that the probability of observing a particle at some future location is proportional to the squared modulus of the amplitude of its wave function. Although this statistical relationship is well quantified, the interpretations have remained controversial, with many split between the classical Copenhagen interpretation and some variation of the de Broglie-Bohm pilot wave models. Recent experiments with Hydrodynamic Quantum Analogs (HQAs) have demonstrated that droplets of real fluid may achieve stable dynamical states, where interaction of the droplets with their own ripples results in motion analogous to the motion of particles subject to the guiding equation under the pilot wave models. Indeed, many effects previously thought to be exclusively quantum have now been observed as emergent phenomena in these macroscopic HQAs. This has motivated us to explore the possibility that quantum mechanics may actually be the result of fluid dynamics on some real quantum scale fluid. In this paper, we show that if there is a real quantum scale fluid having dynamics analogous to the fluid in HQAs, then this fluid must be a superfluid, and the dynamics of that fluid must take place on the surface of a 4-dimensional hypersphere. Under the influence of cosmological inflation, we further show that these bouncing droplets would have the illusion of a property analogous to rest mass, and that the principles of inertia, momentum, mass-energy equivalence, general relativity, the uncertainty principle and the appearance of a time-like dimension can all be derived for droplets as purely emergent phenomena from the fluid dynamics of this system. As such, we believe that this model merits consideration as a potential foundation for a new unifying theory of physics at all scales.