general relativity and special theory of relativity. This demand means that the laws of physics are to be the same in any coordinate system, cf. A point or an event in the spacetime is represented by x = ( cx 0( s), x 1( s), x 2( s), x 3( s)) in some reference frame so that events in the spacetime are parametrized by the proper time s.īefore we consider the HJB equation any further, we need to make sure that when performing stochastic optimal control on spacetimes, the expectation over spacetime is invariant with respect to coordinate transformations between reference frames. Therefore we have a 4-dimensional system that is to be controlled. Normal notation for contravariant and covariant tensors is used throughout, as well as the Einstein summation convention. we are working in the normal spacetime setting. There are three space dimensions and one time dimension, i.e. We work in units so that we choose the reduced Planck’s constant to be \(\hslash =1\). There is also thread of literature, where stochastic quantization is incorporated with Special Theory of Relativity, see for example 9, 10, but analytic continuation is not explained, again. It should also be noted that recently it has been shown that the famous Heisenberg uncertainty relations seem to be inherent to stochastic systems in general, and they are not unique to quantum mechanical systems, see the recent paper 8. Alas, what is missing also from Ohsumi’s paper is a proper and physically meaningful explanation why the Schrödinger equation is the diffusion equation in imaginary time. In terms of more recent research, see the paper by Ohsumi 7. The mathematical apparatus of linear operators yields also other useful tools such as spectral theory, eigenfunction expansions and so forth. Linearity of the Schrödinger equation is important due to the well-known properties of ‘matter’ waves, such as interference and superposition. This study takes the control view as a starting point, where ultimately the Schrödinger equation is essentially the Hamilton-Jacobi-Bellman (HJB) equation from optimal control theory, when one takes into account relativistic coordinate-invariance of the action and demands linearity. Furthermore Rosenbrock and Ding have done quantum mechanics with control theory 6. Among others, Yasue 4 and Papiez 5 have worked with stochastic control and quantum mechanics in the 1980s. The stochastic optimal control approach to quantum mechanics can be traced back to Edward Nelson 3. The same challenges are omnipresent also in the sphere of relativistic quantum mechanics. Even though the Schrödinger equation is adopted as a postulate of Quantum Mechanics in the literature, we argue that it can be derived in a meaningful manner and therefore from a didactical and pedagogical point of view, the postulate approach is not totally satisfying. Especially confusing is the imaginary nature of these differential operators. The postulates of quantum mechanics are just stated, including and in particular the operator substitution rules for energy and momentum. The literature provides many heuristic ways to justify the equation, but most of the heuristics are not completely satisfactory in terms of understanding. The Born rule then gives us the probability density as \(\rho (\), where the asterisk refers to complex conjugation. Linearity is intimately related to the properties of de Broglie ‘matter’ waves. One of the key properties of the Schrödinger equation is that it is a linear, parabolic partial differential equation. For example, this occurs in the N=4 supersymmetric Yang–Mills theory.For vectors we use the notation with bold fonts, so that for example x = ( x, y, z). In very special cases, it may happen when the couplings and the anomalous dimensions do not run at all, so that the theory is scale invariant at all distances and for any value of the coupling. Such a behavior is called an infrared fixed point. Then at long distances the theory becomes scale invariant, and the anomalous dimensions stop running. In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x → λ x where the beta-function vanishes. Number specifying how a quantum operator changes under dilations
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