About degrees of freedom of fermions

In summary: Consider ##u(p)##, the Fourier component of the Dirac field with momentum ##p##. This is a complex four-component spinor, so we might think that it has four independent complex degrees of freedom. But in fact ##u(p)## must obey##(p_\mu \gamma^\mu + m)u(p) = 0##.To see what sort of constraint this is, look at the case ##p = (m, 0, 0, 0)## (the case of a particle at rest). Then this equation looks like##(m\gamma^0 + m) u(p) = 0##.
  • #1
karlzr
131
2
There are something I don't get about the degrees of freedom(dof).
For massive dirac spinor, there are four complex components or 8 dofs. But for electron/position, there are only 4 dofs in total ( electron up &down, position up&down). Does it mean the equation of motion eliminate the other four dofs? I don't think so if KG equation doesn't eliminate any dof.
Actually, if we write the EOM of massive dirac spinor in terms of left and right-handed weyl spinor, right-handed spinor can be expressed in terms of the derivative of left-handed one and vice versa. Does it mean the two helicity spinors weyl spinors are not independent? I hope not, since they represent distinct spins.
Then it comes to majorana spinor. We all know they can be described by only left-hand or right-hand weyl spinor. Two complex components amount to 4 dofs. But obviously there are only two dofs (spin up and down).
 
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  • #2
karlzr said:
There are something I don't get about the degrees of freedom(dof).
For massive dirac spinor, there are four complex components or 8 dofs. But for electron/position, there are only 4 dofs in total ( electron up &down, position up&down). Does it mean the equation of motion eliminate the other four dofs?

Yes. The Dirac equation in momentum space is

##(p_\mu \gamma^\mu + m)u(p) = 0##.

This equation has two independent solutions for ##u(p)##. The other two complex degrees of freedom are forced to be zero.

karlzr said:
I don't think so if KG equation doesn't eliminate any dof.

Note that multiplying the above equation by ##(p_\mu \gamma^\mu - m)## gives

##(p^2 - m^2)u(p) = 0##

which is the Klein-Gordon equation in momentum space. So the Dirac equation is a much stronger constraint than the Klein-Gordon equation: when you impose the Dirac equation, you automatically impose the Klein-Gordon equation PLUS a constraint on the spinor structure.
 
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  • #3
The_Duck said:
Yes. The Dirac equation in momentum space is

##(p_\mu \gamma^\mu + m)u(p) = 0##.

This equation has two independent solutions for ##u(p)##. The other two complex degrees of freedom are forced to be zero.
Is it possible to come to this conclusion from pure algebra. I don't understand which two complex dofs are set to zero. Is it because the left-hand and right-hand components couple in Dirac equation so they are not independent of each other?

The_Duck said:
Note that multiplying the above equation by ##(p_\mu \gamma^\mu - m)## gives

##(p^2 - m^2)u(p) = 0##

which is the Klein-Gordon equation in momentum space. So the Dirac equation is a much stronger constraint than the Klein-Gordon equation: when you impose the Dirac equation, you automatically impose the Klein-Gordon equation PLUS a constraint on the spinor structure.
That makes sense.
 
  • #4
karlzr said:
Is it possible to come to this conclusion from pure algebra. I don't understand which two complex dofs are set to zero.

Consider ##u(p)##, the Fourier component of the Dirac field with momentum ##p##. This is a complex four-component spinor, so we might think that it has four independent complex degrees of freedom. But in fact ##u(p)## must obey

##(p_\mu \gamma^\mu + m)u(p) = 0##

To see what sort of constraint this is, look at the case ##p = (m, 0, 0, 0)## (the case of a particle at rest). Then this equation looks like

##(m\gamma^0 + m) u(p) = 0##

I want to rewrite this as

##2m\frac{1}{2}(1 + \gamma^0)u(p) = 0##

because the matrix ##\frac{1}{2}(1 + \gamma^0)## is a projection matrix that projects onto a two-dimensional subspace of the four-dimensional vector space in which ##u(p)## lives. For example in one possible basis,

[tex]\gamma^0 = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right)[/tex]

so that

[tex]\frac{1}{2}(1 + \gamma^0) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)[/tex]

Therefore in this basis the equation

##2m\frac{1}{2}(1 + \gamma^0)u(p) = 0##

has the effect of setting the first two complex components of ##u(p)## to zero. This is one example of the fact that for any momentum ##p## the matrix

##(p_\mu \gamma^\mu + m)##

is essentially a projection matrix onto a two-dimensional subspace of the four-dimensional spinor space in which ##u(p)## lives, and so the Dirac equation has the effect of projecting out two of the four complex degrees of freedom of ##u(p)##.
 
  • #5
The_Duck said:
This is one example of the fact that for any momentum ##p## the matrix

##(p_\mu \gamma^\mu + m)##

is essentially a projection matrix onto a two-dimensional subspace of the four-dimensional spinor space in which ##u(p)## lives, and so the Dirac equation has the effect of projecting out two of the four complex degrees of freedom of ##u(p)##.

That clears my doubt about fermions. Can this argument be applied to vector field, like photon? It's said the equation of motion projects out one dof. So there are only 3 dofs for massive vector fields (no gauge invariance for massive vector fields).
 
  • #6
what do yo mean by vector fields?
The vector fields are introduced for gauge invariance...
 
  • #7
ChrisVer said:
what do yo mean by vector fields?
The vector fields are introduced for gauge invariance...

But if the vector field is massive, then the longitudinal mode becomes physical and there is no gauge invariance. Mass term violates gauge invariance.
 

What are degrees of freedom of fermions?

Degrees of freedom of fermions refer to the number of ways in which particles can be arranged or have energy distributed among them. In quantum mechanics, fermions are particles that follow the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state at the same time. This results in a restriction of the possible states that fermions can occupy, leading to a limited number of degrees of freedom.

How are degrees of freedom of fermions different from bosons?

Degrees of freedom of fermions and bosons differ in terms of their spin. Fermions have half-integer spin, while bosons have integer spin. This affects the way in which they behave and interact with each other, resulting in different numbers of degrees of freedom. For example, fermions follow the Pauli exclusion principle and have limited degrees of freedom, while bosons can occupy the same quantum state and have a higher number of degrees of freedom.

Why are degrees of freedom important in understanding fermions?

Degrees of freedom are important in understanding fermions because they provide information about the behavior and properties of these particles. The number of degrees of freedom affects the energy and entropy of a system, which can have significant implications in various fields of physics, such as thermodynamics, statistical mechanics, and quantum mechanics.

How do degrees of freedom affect the behavior of fermions in different systems?

The number of degrees of freedom can greatly impact the behavior of fermions in different systems. For example, in a solid material, fermions can be confined and have limited degrees of freedom, resulting in properties such as electrical conductivity and thermal conductivity. In a gas, fermions have more degrees of freedom and can easily move around, resulting in properties such as pressure and temperature.

Can degrees of freedom be changed or manipulated in fermions?

Yes, degrees of freedom can be changed or manipulated in fermions. This can be achieved through various methods, such as changing the temperature, applying a magnetic field, or confining the particles in a specific space. By altering the degrees of freedom, we can observe changes in the behavior and properties of fermions, providing valuable insights into their nature and interactions.

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