Complex Mechanics ~ An Unified Mechanics

       As one turns the scale to the atom level, the behavior of a particle becomes very unpredictable. Quantum mechanics has a better description of microscopic properties than classical mechanics for this scale. It brings us the most precise and successful numerical predictions in the history of science. But a contradiction accompanied with the development of quantum mechanics was brought to light, and has been queried from realists. It is certain that the probabilistic interpretation contravenes the law of causality, and is unable to delineate the fundamental physical process of the universe. According to quantum mechanics, measurements of some properties, such as a particle's momentum for example, can yield a range of possible results with varying probabilities. In other words the objective physical process, once physicists took it for granted to possess definite properties that suitable observations can reveal, is no longer adaptable in the microscopic world.

         The most profound conceptual difficulties of quantum mechanics are those that originated from conflicting non-causations. The compelling results unearth the incompleteness of quantum mechanics revealed by the EPR experiment¹ and the double-slit experiment, where the former lies in the inadmissible awareness of definite position and momentum at the same time; while the latter designates the wave-particle duality, which is the most unusual character of quantum particles. Moreover, this is just the outset of this battle to uncover the covert facts behind quantum mechanics. More quantum strangeness is rooted in the relevance of causation at a microscopic level, such as the uncertainty principle, entanglement, tunneling effect and so on, which have been proposed since. These phenomena profoundly violate the standpoint of human experience to our orderly, causal universe. It seems that the usage of quantum mechanics has been widely accepted while ignoring its essential features. This metaphysical quantum theory perfectly interprets experimental results, however, addresses nothing which approaches the realistic description with regards to the nature of quantum world, and yet becomes the most wizardly of theories.

         The need for a more complete theory is revealed as our understanding of the physical universe has deepened profoundly, and, in particular, as the desiderative exploration of the very beginning of our universe has been carried out. One of the possible theories is the “hidden variable theory” proposed by D. Bohm. An insinuation of invisible variables revises empirical grounds and preserves the causality on the subject of quantum behavior based on a wave conception. It provides a possible sketch of the parentage of “multi-path” methodology, which is an alternative perspective on the issue of a particle’s wave-like property proposed by R. Feynman. His standpoint concurs exactly with all that went before from the numerical predication point of view, but presented in a causal way. It concludes that there are a lot of trajectories between two fixed points, and becomes only one trajectory, the classical one, at the macroscopic level. However, this causal manner emerges from a contradiction of quantum theory, and has not yet been fully understood from the anthropocentric viewpoint. In terms of physical laws in the objective physical process, it appears that an invisible physical strength acts on a particle and draws forth all possible trajectories. Obviously, more questions arise from the deterministic viewpoint to this hidden variable theory, such as: “What is matter wave exactly?” and “Why is there an invisible part which does generate an effect on particles?” and so on. Those puzzling questions could be straightened out if we could visualize this invisible portion in some way. Therefore, the primary research to be addressed next is how to convert hidden variables into realistic physical quantities and to provide a concrete depiction that can bring a convincing theory forth, which would account for all wired quantum phenomena and describe every property of the quantum world.   

   The main purpose of this introduction is to explore the process of visualizing hidden variables, and to represent a complete theory within microscopic physics. One can imagine that there is a bee in a house with no window. The bee has a special power allowing it to pass through walls to go outside. Of course, we cannot see the bee while it stays outside the house and we are inside. It can only be seen after passing through the wall and coming back into the house again. Hence, what we can observe is that the bee appears all of a sudden and disappears later if it passes in and out of the house. In such a condition, we have no idea about when it will come back to the house for the reason that we cannot see anything outside the house, but what we can do is to estimate the probability of being stung by the bee according to the position we are in inside the house. This is the probabilistic interpretation proposed by quantum mechanics to describe a quantum system.

On the other hand, let us replace all the walls of the house by transparent glass then there will be no doubt that we can see everything outside the house as we stay inside. Now, we still can observe where the bee is and even how it moves after it is passing through the glass wall to go outside. There is no problem for us to predict its flight path, position, velocity and heading. In other words, we can be told when and where the bee comes back into the house in a deterministic way, without estimating the stinging probability. A contrast can be made here that transparent walls symbolize the visualization process; it can reveal motions of the bee outside the house in the former case in an objective physical process. Consequently, a continuous and deterministic interpretation of the quantum world can arise if we can find some method of replacing the invisible border.

It is straightforward to think of extending dimension to bring transparent walls into existence. To deliberate on the imperceptible part to the sense at quantum level, a rational speculation of complex domain could strike a bargain, in which its imaginary part can represent the invisible world. In fact, it is not an unrestrained attempt originated from an intuition only. The complex concept was objective in Schrodinger’s equation and can be made aware of by the appearance of the imaginary factor “i”, and has been permitted as a genuine mathematical tool. In fact, the ignoring of the imaginary sign in wave mechanics can be attributed to practical experimental results, which cause people’s attention to delve into the atom scale. Owing to the limited observable dimension, imaginary features of nature which could dissolve the consequence of experiments has been eliminated by empiricists. However, no thorough canvass can be addressed if our understanding of nature contains a one-legged version of the full view. This is the main reason why quantum mechanics is an incomplete theory, for its grounds for existence lie on observation that has been criticized from the philosophical aspect and the causality of its nature.                   

In pioneering work approaching causal quantum physics, a remarkable achievement based on the complex concept has been proposed by C. D. Yang. In his study of the physical process in a complex domain, astonishing results have been acquired. They reveal that Schrodinger’s equation is a deformation of the Hamilton-Jacobi equation when considering the existence of a complex dimension. There is no fortune at all in this since it is the only way to manifest the true face of its nature after a long struggle for a unified description of the causal universe up to now. General relativity and causal quantum physics can for the first time be on an equal footing in the foundation of the Hamiltonian outlook on causal physical processes. The biggest difference of Hamiltonian in the microscopic world is its additional term, namely quantum potential, as compared to the classical one. It states that there is a special field in the atom scale. It is on the decrease along with increasing mass, and finally vanishes in our daily scale. This field is so-called the quantum field, which can take responsibility for all marvelous quantum phenomena from the perspective of causality. Thus, a concrete object having specific physical quantities can be discussed after a complete description of carried energy has been expressed. In other words, it can be regarded as a classical particle for those who once considered it as the most bizarre particle in the quantum scale. 

One of the most elusive parts of this causal quantum physics, or so-called quantum Hamilton mechanics, is the existence of complex dimension. It becomes an objectionable point for those who only can be convinced through measurements. In reality, there is an objective world whose nature and reality are independent of human observers. Hence, if the projection of causality on our living world, given by the complex dimension extension, can bring us a compatible result with what we can observe, then, it could be considered to bring complex dimension into existence. Unlike the quantum potential proposed in a hidden variable theory, the quantum potential we discuss in complex space can bring up compatible outcomes with a quantum probabilistic interpretation. The unpersuasive exposition of motion in an eigen state once becomes a fatal wound as a causal theory in Bohmian mechanics, however, becomes describable in quantum Hamilton mechanics. It is reasonable to explain the multi-path behavior by thinking of no specific initial position that has been made, since its imaginary initial position cannot be confirmed as the real one has been fixed in an experimental process. In addition, a particle is moving in a complex domain when a complex force associated with the quantum potential acts on it. As a result, a non-classical trajectory, which has deviations from the classical straight line, can be observed in the double-slit experiment issue. After a long period, the ensemble of trajectories presents us with the formation of a wave, which has been regarded as the matter wave, exhibited by the interference result of the dark and bright band on the screen. It is the best way to illustrate the on-again-off-again strange property of a quantum particle by extending dimension to a complex domain. In such a way, we can see everything through the transparent wall, and can explore all unexplainable and unpredictable quantum strangeness.

Therefore, the wave function decided by Schrodinger’s equation describes a particle’s motion statistically and cannot provide more detail about each trajectory. It is clearer to think of it as a water flow; since we cannot know a specific molecule’s motion by observing its whole flow, and can only understand the probability of this molecule passing by a specific area. This is the limitation of describing a quantum motion based on wave mechanics since it provides a macroscopic observation which cannot be overlooked. On the contrary, a fully informative view of a particle’s motion can be presented in terms of causal description from the same wave function. We can observe a specific particle’s motion with the help of quantum Hamilton mechanics since all its moving information can be traced in complex space. In other words, the incompleteness of a quantum mechanical description can be regarded as its macroscopic observation and becomes a functional tool that can only extract a portion of information from real conditions. Moreover, wave description itself has constraints on the local and accuracy standpoints of quantum motions. On the subject of the EPR problem, the non-local property can be explained by the information propagation through the quantum field between two particles. And the uncertainty principle can be regarded as the consequence of statistical observation. Now, we have a logically consistent and empirically adequate deterministic theory of quantum phenomena.     

In summary, the revolutionary viewpoint of the complex dimension not only visualizes hidden variables but also provides them with a realistic physical meaning. As a causal theory, quantum Hamilton mechanics can be fully understood based on the classical standpoint, and invigorates complete description of the microscopic nature. However, this complex extension should be amenable to revision on empirical gounds as a scientific hypothesis. The prediction of small perturbation of the electron's spin momentum can be one of the crucial testmonies, waiting for the comparison with experiments. On the other hand, the flying time of a photonic tunneling effect can be brought out by solving the equation of motion of the photon, which can be examined by the experimental data. The oscillation period of ammonia molecular can be estimated precisely by averaging the time that a trajectory cycle has made. Those issues have been executed in our laboratory recently, and could be worked out in the near future. We can conclude that quantum mechanics is a theory of phenomenology which gives average values for observed quantities; while quantum Hamilton mechanics is a theory of fatalism presenting specific particles and trajectories. Quantum Hamilton mechanics as a causal theory extends our understanding of the true nature of the microscopic world. It defends the causality, preserves physical laws while revealing the most mysterious feature of nature, indicating what is behind quantum mechanics.