Categories
- Art (356)
- Other (3,632)
- Philosophy (2,814)
- Psychology (4,018)
- Society (1,010)
Recent Questions
- Why did everyone start to hate the Russians if the U.S. did the same thing in Afghanistan, Iraq?
- What needs to be corrected in the management of Russia first?
- Why did Blaise Pascal become a religious man at the end of his life?
- How do I know if a guy likes you?
- When they say "one generation", how many do they mean?
First, read this part about calibration particles. To understand what will be written next.
An electron is a Dirac particle – that is, it is described by the Dirac equation. This equation has an internal symmetry with respect to transformations by elements of the group U(1). These elements are represented as complex exponentials. If you multiply the solutions of the Dirac equation by complex exponentials, these solutions will remain solutions of the Dirac equation if the arguments of these exponentials do not depend on coordinates and time (that is, in the case of global calibration). This symmetry, according to Noether's theorem, leads to the conservation of the electron's electric charge. But when we take causality into account , we are forced to make the exponent argument depend on coordinates and time. Because otherwise, all the fields in the universe will be calibrated (multiplied by our exponent) simultaneously. And according to the special theory of relativity, these changes cannot occur instantly – but must propagate at the maximum speed of light. Given the dependence of the exponent argument on coordinates and time, we just take into account the instantaneous propagation of the calibration. This calibration is called local. In this case, the electron itself, being a solution of the Dirac equation before performing local calibration, ceases to be its solution after the product of local calibration, and becomes the solution of some new equation describing this electron + another field.
It can be shown that this field is exactly an electromagnetic field, and its quanta are photons.
So, due to the presence of U(1) symmetry in the equation describing the electron and any other charged particles, their charge is firstly preserved, and secondly, these charged particles are always connected to the electromagnetic field – this is such a “fur coat” of photons.
Particles that do not have this symmetry are uncharged and do not have a “coat” of photons around them.
Thus, the electron is connected to the photon through the internal symmetry of the equation describing the electron.
More detailed description for those interested:
Any electromagnetically charged particles interact with each other through carriers – called photons.
Where does this photon come from anyway? And it is taken from a kind of “fur coat” of the electromagnetic field in which any charged particle is dressed. If we take the basic relation between the energy and momentum of particles from the special theory of relativity (E ^ 2 – P^2 = m^2) and quantize it, we can get two equations – the Klein-Gordon equation describing bosons. And the Dirac equation describing fermions.
A digression from the topic: (At this point, by the way, in a natural way from the basic postulates of quantum mechanics, not postulating the presence of spin in any way, but using only special relativity, particles have such a quantum number as spin. This characteristic has no classical analogues and appears from the symmetry of the Klein-Gordon and Dirac equations with respect to rotations of the coordinate system. It appears as another conserved quantity. It's just that in a more correct theory, it's not the angular momentum that is conserved, but the sum of the angular momentum and spin. Angular momentum is generally not conserved in cases where classical mechanics speaks of its conservation. Well that's a bit of a digression)
But the Dirac and Klein-Gordon equations themselves do not quite correctly describe unobservable quantities. Because if we know that the solution is an unobservable quantity, then there must be transformations of these solutions that leave these solutions as solutions of the original equations. Because otherwise everything will change and we will be able to see it (solutions become observable). In field theory, unobservability is associated with the inability to determine the absolute value of the potential – we can only measure the potential difference. This means that adding a constant to the solution (potential) should not change anything. In quantum mechanics, unobservability is related to the ability to rotate solutions in the complex plane (sorry for being so abstract), that is, to multiply solutions by complex exponentials. In quantum field theory (the synthesis of quantum mechanics and special relativity), solutions of the Dirac and Klein-Gordon equations (K-R) have both symmetries. The transformations described above are called calibrations (global calibrations). But these symmetries exist only for the time being, until we understand that there is a special theory of relativity (SRT). And so if, for example, there is a certain field in the entire space, then according to SRT it cannot be calibrated instantly-as if… any such calibration initiated at a given point in space must propagate along the entire field at a maximum of the speed of light. Mathematically, this is expressed as the dependence of the calibration transformations on coordinates and time. Such calibrations are called local calibrations.
And if we take into account the instantaneous spread of calibration, then horror! With such transformations of solutions, they cease to be solutions of the original Dirac and K-R equations! That is, they become observable. The whole theory is broken! But if you take a closer look, let our solutions cease to be solutions of the Dirac or K-G equations and become solutions of some other equation (2). Now, if you calibrate the solutions again , you can see that they remain solutions of equation (2). And for the third time, and for the fourth, and for the 44th, and so on… It seems that equation (2) is more correct. In addition, it has all the properties of the original equations, only now it also correctly describes the unobservable ones. What is this equation 2? And this is a quantized equation of classical field theory describing the interaction of a charged particle with an electromagnetic field! From this it is concluded that any charged particle is dressed in a kind of fur coat made of an electromagnetic field. Naked charges not wearing a fur coat from the electromagnetic field do not exist in nature. After quantization, the quanta of this field will be the same photons. They belong to the class of calibration particles because they are obtained from calibration. They are obtained as vectors, since the solutions of the classical field theory equation themselves were vectors (vector potentials) before quantization. In general, the resulting equation (2) is the original Klein-Gordon or Dirac equation + one more component of the equation. This component is called an interaction component. And it can be considered by itself separately. Because in the general case, it would be possible to conduct similar reasoning with equations describing some other charged particles and get the same interaction component. The interaction component itself is a K-G equation. So its solutions (photons) are bosons. In general, the described symmetry is called U(1) – it is a calibration by multiplying solutions by complex exponentials. All electromagnetically charged particles and only they are characterized by the presence of such a symmetry in the equation describing these particles. Therefore, photons interact only between charged particles and no others. The photon equation itself says that the photon has no charge.