ψ in formula for strong force currents, how many components?

In summary, "ψ" is used to represent the wave function of a particle in the formula for strong force currents in quantum mechanics. It is used in the Schrödinger equation to calculate the probability of a particle's interaction with the strong force. The wave function can have multiple components or "spin" states, which can affect the calculation. Other factors such as particle type, energy, and distance also play a role in the calculation of strong force currents. The study of strong force currents is crucial in understanding the behavior of subatomic particles and the fundamental building blocks of the universe.
  • #1
Spinnor
Gold Member
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In the ψ below, there are 4 components for the Dirac spinnor times three possible color states for a total of 12 components for ψ?

ChrisVer said:
...

Nevermind, to get the color current, you need the interactive Lagrangian:

[itex]L_{int}= -g_{3} \bar{ψ} γ^{μ}λ^{a} ψ A_{μ}^{a}/2 [/itex]

the corresponding conserved current (if you remember from the Dirac's current case) is:
[itex] J_{SU(3)}^{μa}= g_{3} \bar{ψ} γ^{μ}(λ^{a}/2) ψ[/itex]

What can we see from that? That we have 8 conserved currents. Each of them is individually conserved. The continuity relation for the currents, is given by their conservation, thus you have again 8 different continuity relations:
[itex]∂_{μ}J_{SU(3)}^{μa}= 0 [/itex]

and the color charge is:
[itex] Q_{c}=\int{d^{3}x J_{SU(3)}^{0a}}[/itex]


If they also carry electric charge, you'll get also another current, corresponding to [itex] U(1)_{Q} [/itex] interaction...

[/itex]

...

Are there low energy, weak field limits of the above that allow us to consider classical color counterparts of electric current densities and electric charge densities?

Thanks for any help!

Thanks to ChrisVer for the original post!
 
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  • #2
No, color confinement prevents anything like a classical color current.
 
  • #3
Spinnor said:
In the ψ below, there are 4 components for the Dirac spinnor times three possible color states for a total of 12 components for ψ?

Yes. Unfortunately in QFT we are usually swimming in a sea of suppressed indices. Here the Dirac and color indices of the ##\psi## field have been suppressed. If you write out all the indices in the expression for the SU(3) current, you get

##j^{a \mu} = \bar \psi_{\alpha i} \gamma^\mu_{\alpha \beta} \lambda^a_{ij} \psi_{\beta j}##

Here ##\alpha## and ##\beta## are Dirac indices, which take on four possible values; ##i## and ##j## are indices for the fundamental representation of SU(3), and take on three possible values; ##a## is an index for the adjoint representation of SU(3), and takes on eight possible values; and ##\mu## is a Lorentz index and takes on four possible values.

##j^{a0}## is the color charge density and ##j^{a\mu}##, ##\mu = 1, 2, 3## is the color current density.
 
  • #4
dauto said:
No, color confinement prevents anything like a classical color current.

I think it was shown to me that asymptotic freedom results because the number of quark types is less then 16. Could we arbitrarily add more quark types to the standard model so as to obtain my classical weak field low energy color physics limit?

At high energies in a quark/gluon plasma does need for colorless combinations of quarks and antiquarks still apply?

Thanks for the help!
 
  • #5
The_Duck said:
Yes. Unfortunately in QFT we are usually swimming in a sea of suppressed indices. Here the Dirac and color indices of the ##\psi## field have been suppressed. If you write out all the indices in the expression for the SU(3) current, you get

##j^{a \mu} = \bar \psi_{\alpha i} \gamma^\mu_{\alpha \beta} \lambda^a_{ij} \psi_{\beta j}##

Here ##\alpha## and ##\beta## are Dirac indices, which take on four possible values; ##i## and ##j## are indices for the fundamental representation of SU(3), and take on three possible values; ##a## is an index for the adjoint representation of SU(3), and takes on eight possible values; and ##\mu## is a Lorentz index and takes on four possible values.

##j^{a0}## is the color charge density and ##j^{a\mu}##, ##\mu = 1, 2, 3## is the color current density.


Can the above sum be simply realized as a single multicomponent matrix sandwiched between two multicomponent vectors?

Thanks for your help!
 
  • #6
Spinnor said:
Can the above sum be simply realized as a single multicomponent matrix sandwiched between two multicomponent vectors?

Thanks for your help!

Sure, if you like you can think of ##\psi## and ##\bar \psi## as 12-component vectors and ##\gamma^\mu \lambda^a## as a ##12 \times 12## matrix.
 
  • #7
Is that true Duck? I mean it'd be like breaking the tensor product of representations into tensor sum
 
  • #8
The_Duck said:
Yes. Unfortunately in QFT we are usually swimming in a sea of suppressed indices. Here the Dirac and color indices of the ##\psi## field have been suppressed. If you write out all the indices in the expression for the SU(3) current, you get

##j^{a \mu} = \bar \psi_{\alpha i} \gamma^\mu_{\alpha \beta} \lambda^a_{ij} \psi_{\beta j}##

Here ##\alpha## and ##\beta## are Dirac indices, which take on four possible values; ##i## and ##j## are indices for the fundamental representation of SU(3), and take on three possible values; ##a## is an index for the adjoint representation of SU(3), and takes on eight possible values; and ##\mu## is a Lorentz index and takes on four possible values.

##j^{a0}## is the color charge density and ##j^{a\mu}##, ##\mu = 1, 2, 3## is the color current density.

And that's just for a single quark. For the complete current you have to sum over all quarks which means two more indices (one for the isospin and one for the family).
 
  • #9
ChrisVer said:
Is that true Duck? I mean it'd be like breaking the tensor product of representations into tensor sum

This isn't about breaking up a tensor product into a tensor sum; it's just about how you can write a member of a tensor product representation as one big matrix. As a concrete example, consider a system of two spin-1/2 particles. It lives in the tensor product representation ##\frac{1}{2} \otimes \frac{1}{2}## of the rotation group. The representation has dimension four and a convenient choice of basis states is ##|\uparrow\uparrow\rangle, |\uparrow\downarrow\rangle, |\downarrow\uparrow\rangle, |\downarrow\downarrow\rangle##. Then on this basis we can write the tensor product operators as single ##4 \times 4## matrices, for example

[tex]\sigma_3 \otimes \sigma_3 = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)[/tex]

[tex]\sigma_1 \otimes \sigma_1 = \left(\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)[/tex]

Similarly ##\gamma^\mu \lambda^a## can in principle be thought of as a ##12 \times 12## matrix acting on the dimension-12 tensor product representation that ##\psi## lives in.
 
  • #10
what happens to electrons in nuclear reaction ?
 

1. What is the meaning of "ψ" in the formula for strong force currents?

"ψ" represents the wave function of a particle and is used to describe the probability of a particle's position or momentum in quantum mechanics. In the context of strong force currents, it represents the probability amplitude of the particle's interaction with the strong force.

2. How is the wave function used to calculate strong force currents?

The wave function is used in the Schrödinger equation, which describes how the wave function evolves over time. By solving this equation, we can determine the probability of a particle interacting with the strong force at a particular time and location.

3. Can the wave function for strong force currents have multiple components?

Yes, the wave function can have multiple components, known as its "spin". This is because the strong force interacts with particles of different spin orientations in different ways. For example, protons and neutrons have different spin states and therefore have different wave functions when interacting with the strong force.

4. Are there any other factors that affect the calculation of strong force currents?

Yes, in addition to the wave function, other factors such as the type of particle, its energy, and the distance between particles can also affect the calculation of strong force currents. These factors are taken into account in the Schrödinger equation and other quantum mechanical models.

5. How does the calculation of strong force currents relate to the behavior of subatomic particles?

The calculation of strong force currents is essential in understanding the behavior of subatomic particles, such as protons and neutrons. These particles are held together by the strong force, and their interactions with this force can be described by the wave function and other quantum mechanical models. By studying the behavior of strong force currents, we can gain a deeper understanding of the fundamental building blocks of the universe.

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