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Editorial Reviews. Review. For all readers interested in specific aspects of perturbation Introduction to Quantum Mechanics:Schrödinger Equation and Path Integral - Kindle edition by Harald J W Müller-Kirsten. Download it once and read it.

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About Harald J. Harald J. Books by Harald J. Trivia About Introduction to Q No trivia or quizzes yet. Some researchers still think the many worlds interpretation of quantum theory exists only to the extent that the associated basis problem is solved [ 4 , 5 , 6 ].

Many-minds interpretation is the extension of many worlds interpretation. The thought of this interpretation is when an observer measures a quantum system, then a state that is consistent with minds which produced by the observer brain, called mental states, will entangle with this quantum system. The mental state of the brain corresponding with this system is involving, and ultimately, only one mind is experienced, leading the others to branch off and become inaccessible. In this way, every sentient being is attributed with an infinity of minds, whose prevalence corresponds to the amplitude of the wave function.

As an observer checks a measurement, the probability of realizing a specific measurement directly correlates to the number of minds they have where they see that measurement. However, like the many worlds interpretation, the many-minds interpretation is still a local theory. Additionally, it tosses the basis-preferred problem to the mentality of observer and makes this physical problem fall into an endless discussion of mental state of human.

It is proposed by Bassia and Ghirardia [ 10 ]. However, it is still a nonrelativistic theory and remains the nonlocality problem. However, these three expressions have their own focuses. On the other hand, in quantum mechanics, when do a measurement on a wave function diffusing in all of space, such as the measurement of the position of an electron in the experiment of double-slit interference, we will find that the whole wave function will instantaneously collapse to this position measured with some probabilities.

Moreover, we notice that the action integral in Feynman path integral formulation is the classical form. The classical physics is born to be a local theory and of course cannot exhibit the character of nonlocality.

However, the relativity theory is different. In relativity theory, the time and space are coupling. Beyond the light cones in Minkowski space, the space-time causality is broken, and this may cause the nonlocality. The superluminal velocities are forbidden in real world, but for a connection description of virtual paths in the path integral theory, it might be practicable. What will happen when we extend the classical action to relativistic action?

Could the superluminal trajectories included in possible paths to calculate quantum amplitude in the Feynman theory cause the nonlocality? These questions will be revealed when we extend the Feynman path integral. This assumption makes Feynman path integral very successful in nonrelativistic quantum theory, but it is also the top offender that impedes the integration between Feynman path integral and relativity in non-field theory. Why should this be?

## Get e-book Introduction to Quantum Mechanics:Schrödinger Equation and Path Integral

For the extension, it is necessary to break up this assumption, and Eq. It should be transformed into Feynman path integral in low-energy and low-velocity condition. Under these four limitations, the forms of R and W p are very few.

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The final forms of R and W p chosen in extended Feynman path integral are. This means Eq. What is concerning then for us is what we can get from Eq. It is hard to directly calculate the value of Eq. A way to get some results from Eq.

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In this process:. The integral form should be departed into two parts: the part that contains the low-velocity paths and the part that contains superluminal-velocity paths:. This can be exactly calculated. In the following context, we will detail this calculation in 1D space for simplification. The methods of the calculation in 2D and 3D are the same. Similarly, we can also get the expression of I 0 :. The contour integral is used in the last step as shown in Figure 1.

Contour integral. This figure shows the contour integral in a complex plane. For this contour integral, there is no singular point, and of course the total integral value is zero. Integrating Eq. Summation in Eq. And Eq.

## Gestione Didattica - Politecnico di Torino

It is more complicated to get Eq. The detailed deduction can be seen in supplementary online material of the reference [ 13 ]. It should be mentioned that Eq. To construct a Lorentz covariant, the antiparticle wave function should be introduced.

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Combining Eqs. In , Oskar Klein and Walter Gordon proposed this relativistic wave equation. However, it was found later that this equation is not suitable for one particle because the probability density is not a positive quantity, which means the particle can be created and annihilated arbitrarily in Klein-Gordon equation [ 14 ]. The extended Feynman path integral shows the explanation for this non-positive probability density here. The wave function that is determined by Klein-Gordon equation is the mixed state of the particle and its antiparticle. Because particles and antiparticles can be annihilated each other to a vacuum state, and the vacuum state can produce particles and antiparticles, so the mixed state with superposition state of a particle and an antiparticle is a matter of course of a non-positive quantity.

In quantum mechanics, the continuity equation describes the conservation of probability density in the transport process. It is a local form of conservation laws. It says the probability cannot be created or annihilated and, at the same time, also cannot be teleported from one place to another.

However, in the extended Feynman path integral, the density-flux equation will be revised, and the local conservation is broken. In extended Feynman path integral, the density-flux equation can be written as the following formula:.

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The last term in the right of Eq. From the theory of Neumann, the difficulties of understanding collapse are the probability, which seems incompatible with the deterministic time-evolution equation, and the instantaneity, which seems that it breaks the special relativity theory. In this section, we will show that these puzzling characters are due to the potential noise and nonlocal correlation or relativistic effect.

Let us return to Eq. The superluminal paths are included when we calculate the propagator. The superluminal paths will support complex phases in Eq. These complex phases are the main culprits that cause the nonlocal correlation. To describe this mechanism concisely, the nonlocal correlation produced in 1D space is just detailed here. Assume a system in the potential field with the scalar potential U x and vector A 0 x.