This document contains comments about the article "Relativistic quantum mechanics" in Wikipedia
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.
Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity, but not general relativity.
Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators
There exists no Galilean relativity.
Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity, but not general relativity.

### 1. Combining special relativity and quantum mechanics

A fundamental prediction of special relativity is the relativistic energyñmomentum relation; for a particle of rest mass m, and in a particular frame of reference with energy E and 3-momentum p with magnitude in terms of the dot product p = sqrt(p.p), it is
E^2 = c^2 p.p + ( m * c^2 )^2 .
The biggest problem is how do you calculate the parameters m and v in reality in a experiment.
These equations are used together with the energy and momentum operators, which are respectively:
E^ = etc , p ^ = etc ,
to construct a relativistic wave equation (RWE): a partial differential equation consistent with the energyñmomentum relation, and is solved for psi to predict the quantum dynamics of the particle.
First of all you have to calculate all the parameters based on experiments!
For space and time to be placed on equal footing, as in relativity, the orders of space and time partial derivatives should be equal, and ideally as low as possible, so that no initial values of the derivatives need to be specified.
This is a very tricky sentence, specific the final part. If the correct order is two than you should not try one because that is wrong!
This is important for probability interpretations, exemplified below. The lowest possible order of any differential equation is the first (zeroth order derivatives would not form a differential equation).
All (?) the probabilities arrive because specific at atomic or subatomic level repeating the same experiment always shows different results.

### 1.1 Space and time

In classical mechanics and non-relativistic QM, time is an absolute quantity all observers and particles can always agree on, "ticking away" in the background independent of space.
This silently assumes that what each observer observes, is not the reality, but requires a certain form of mathematics i.e. transformations, to calculate the reality, completly independent of the observer. These transformations often, involve the speed of light.
In relativistic mechanics, the spatial coordinates and coordinate time are not absolute; any two observers moving relative to each other can measure different locations and times of events.
In classical mechanics this is the same but the transformation rules will resolve this to describe an universe which is absolute.
The position and time coordinates combine naturally into a four-dimensional spacetime position X = (ct, r) corresponding to events, and the energy and 3-momentum combine naturally into the four momentum P = (E/c, p) of a dynamic particle, as measured in some reference frame, change according to a Lorentz transformation as one measures in a different frame boosted and/or rotated relative the original frame in consideration.
All of this seems much more complex as in the case of classical mechanics.
What should be agreed upon is that in both theories the original observations and the final predicted observations, describing a situation in the future, should match the future observations. If the mathematics used, to make these predictions, is too complex that no solution is avaible, then a simpler strategy is better.
A typical sentence is the following:
The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.
Under a proper orthochronous Lorentz transformation (r, t) --> delta(r, t) in Minkowski space, all one-particle quantum states psi,sigma locally transform under some representation D of the Lorentz group: psi,sigma(r,t) becomes etc.

### 1.2 Non-relativistic and relativistic Hamiltonians

A key difference is that relativistic Hamiltonians contain spin operators in the form of matrices, in which the matrix multiplication runs over the spin index sigma, so in general a relativistic Hamiltonian:
H^ = H^ ( r, t, p^ , S^ )
is a function of space, time, and the momentum and spin operators.
This makes its all the more difficult to calculate such an Hamiltonian based on experiments.

### 6 History

SR, found at the turn of the 20th century, was found to be a necessary component, leading to unification: RQM.

### 6.1 Relativistic description of particles in quantum phenomena

IMO in all the example the word relativistic is a misnomer.

### 6.2 Experiments

IMO in all the examples SR is not an issue.
1958 Discovery of the Mˆssbauer effect: resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid, useful for accurate measurements of gravitational redshift and time dilation, and in the analysis of nuclear electromagnetic moments in hyperfine interactions.
The concept of time dilation in this specific case requires more study. Wikipedia: https://en.wikipedia.org/wiki/M%C3%B6ssbauer_effect does not mention this. Also there is nothing mentioned about QM.

### 6.3 Quantum non-locality and relativistic locality

In 1935; Einstein, Rosen, Podolsky published a paper concerning quantum entanglement of particles, questioning quantum nonlocality and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances.
The concept of entanglement (i.e. correlation) does not require instantaneous action at a distance. The cause of the entanglement is in the common source or reaction at the instant when the particles are created.
This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by two observers (one observer has to send a signal to the other, which cannot exceed c)
There is no information transmission involved. All the information comes from all the previous identical experiments performed, as such both observers know that there is a correlation.
In 1964, Bell's theorem was published in a paper on the EPR paradox, showing that QM cannot be derived from local hidden variable theories if locality is to be maintained.
This is a typical sentence which cannot be declared right or wrong because it uses to many concepts which are not clear. IMO the only proper way to do science is by performing experiments.

### Reflection 1: Relativistic Quantum Mechanics

Relativity in its basics, is a reaction towards Newton's Law, to describe the movement of objects more accurate.
Relativity consists of 2 parts: Special relativity and General relativity.
• Special Relativity is specific related towards time and length.
• General Relativity is more related towards moving objects and mass

For SR time and length are not considered constants but involve change as a function of the speed of light.
The basic idea is an observer at rest with has a clock which shows rest time and with a rod which has a rest length. From his point of view a moving clock undergoes time dilution and a moving rod length contraction as described by the Lorentz transformations.
Mass in relation towards SR also uses the same mathematics. Mass also has a rest mass and a moving mass changes as described by the Lorentz transformations.

This causes a problem related to for example planet and star sized objects. How do you calculate the rest mass of each while they are always in movement?
The same problem exists also for the elementary particles. IMO it does not make sense to assume a rest mass, while any a sub-atomic particle is never at rest.

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Created: 1 February 2017

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