Question about lorentz-covariance of Dirac equation

In summary: This is important, because in the rest frame of an observer, the value of the field at any particular point is a function only of the value of the field at the observer's rest frame. Now let's boost an electron up to some high energy. In the rest frame of the electron, its field has a value of 1. However, from the perspective of an observer who is watching the electron from far away, the value of the field is not 1. In particular, if the observer boosts the electron up to a momentum p, the value of the field at that point is not just 1+p, but rather 1+p*(1+p). This is where the Lorentz transformation
  • #1
aiqun
1
0
ψ(x) is the four-component wave function of the Dirac equation,that is ψ(x) can be expressed by a column vector (ψ1(x) ψ2(x) ψ3(x) ψ4(x)) ,under a lorentz transformation,it will become ψ'(x').I am confused that how ψ'(x') can be expressd
in the form which is stated by textbooks: ψ'(x')=S(a)ψ(x)
(a is the matrix of the lorentz transformation ,S(a) is a 4*4 matrix which is a function of the parameters of a )
clearly.ψ'(x') is a funtion of x',ψ(x) is funtion of x,my question is for example ,how can
ψ'1(x') can be expressed by a linear combination of ψ1(x), ψ2(x) ,ψ3(x)and ψ4(x)?
is there someone can help me?
 
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  • #2
try just to do it for some special cases, e.g.

start from the dirac-plane waves at rest, and boost them to a momentum p

etc
 
  • #3
First let's do a scalar field. A classical real scalar field assigns a real number to each point in spacetime. All observers agree on the value of that number for each point. If Alice uses coordinates x, she can write down a function [itex]\varphi(x)[/itex] that gives the number assigned by the field to the point she labels with coordinates x. Bob uses different coordinates x', related to Alice's by a Lorentz transformation, x'=ax. Bob also uses a different function, [itex]\varphi'[/itex], of his coordinates. However, since Bob and Alice agree on the value assigned by the field to any particular point, the numerical values of Bob's function of Bob's coordinates must agree with the numerical values of Alice's function of Alice's coordinates; that is, we must have [itex]\varphi'(x')=\varphi(x)[/itex].

For fields in other representations of the Lorentz group, such as a Dirac field, this gets generalized to [itex]\psi'(x')=S(a)\psi(x)[/itex], where S(a) is a matrix that acts on the index carried by the field.
 

1. What is the Lorentz-covariance of the Dirac equation?

The Lorentz-covariance of the Dirac equation refers to the fact that it remains unchanged under Lorentz transformations, which are mathematical transformations that describe the relationship between space and time in special relativity.

2. Why is Lorentz-covariance important in the context of the Dirac equation?

Lorentz-covariance is important because it allows us to apply the Dirac equation to any inertial reference frame, regardless of its relative motion. This is essential in understanding the behavior of particles at high speeds, as described by special relativity.

3. How does the Dirac equation differ from other quantum mechanics equations in terms of Lorentz-covariance?

The Dirac equation is unique in its ability to incorporate both special relativity and quantum mechanics. Unlike other quantum mechanics equations, it is Lorentz-covariant, meaning it can accurately describe the behavior of particles at relativistic speeds.

4. Can you explain the mathematical concept of Lorentz-covariance and how it applies to the Dirac equation?

Lorentz-covariance is a mathematical concept that describes the symmetry of physical laws under Lorentz transformations. In the context of the Dirac equation, this means that the equation remains the same regardless of the inertial reference frame in which it is observed.

5. How is Lorentz-covariance related to the principle of relativity?

The principle of relativity states that the laws of physics should be the same for all observers in inertial frames of reference. Lorentz-covariance is a mathematical manifestation of this principle, as it ensures that the Dirac equation remains consistent and accurate in all inertial frames of reference.

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