SU(2) Spinor: Is Product of Two Scalar-Type Entity?

In summary, the product of two spinors in SU(2) is a scalar and can be treated like a scalar, meaning it can be moved around without worrying about order. The commutativity of the product depends on whether the spinors are fermionic or bosonic, but in this case of two squared spinors, they are bosonic and can commute with each other. However, if they are operators instead of classical fields, caution is needed.
  • #1
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I'm having a memory blank on this particular area of field theory. Is the product of two spinors a scalar or scalar type entity and if so, can I treat it like a scalar? (i.e. move it around without worrying about order etc)

i.e.

is [tex] \Phi_1^{\dagger} \Phi_1 [/tex] a scalar?

and if so does:

[tex] \Phi_2 \left(\Phi_1^{\dagger} \Phi_1\right) = \left(\Phi_1^{\dagger} \Phi_1\right) \Phi_2 [/tex]

where both phi's are SU(2) complex spinors.

Thanks
 
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  • #2
Yes, if you think back to the definition of SU(2), it is precisely the group under which quantities like [itex]\psi^\dagger \psi[/itex] are scalars (namely, it preserves a hermitian inner product). Also, whether you can commute those quantities is not related to whether they are scalars, but does depend on if they are fermionic or bosonic (though, in this case, anything squared is bosonic, so commutes with everything). Also, if they are operators rather than classical fields you need to be careful.
 
  • #3
for your question. The product of two spinors in SU(2) field theory is not a scalar, but rather a complex spinor. This means that it has both a magnitude and a direction, similar to a vector. Therefore, you cannot treat it like a scalar and move it around without worrying about order. The product of two spinors is not commutative, so the order in which you multiply them matters.

In terms of your specific example, \Phi_1^{\dagger} \Phi_1 is a complex number, but \Phi_2 \left(\Phi_1^{\dagger} \Phi_1\right) and \left(\Phi_1^{\dagger} \Phi_1\right) \Phi_2 are not equivalent. This is because \Phi_2 is a complex spinor and the order of multiplication matters.

I hope this helps clarify the nature of the product of two spinors in SU(2) field theory. It is important to keep in mind that spinors are a unique mathematical object and cannot be treated exactly like scalars or vectors. If you have any further questions, please don't hesitate to ask.
 

1. What is SU(2) Spinor and how is it related to scalar-type entities?

SU(2) Spinor is a mathematical representation of a fundamental symmetry group in quantum mechanics. It is related to scalar-type entities because it describes the transformation properties of particles with spin 1/2, which are considered scalar-type entities in the context of quantum mechanics.

2. How does the product of two scalar-type entities in SU(2) Spinor work?

The product of two scalar-type entities in SU(2) Spinor is a mathematical operation that combines the transformation properties of two particles with spin 1/2. This operation is necessary to describe the behavior of composite particles, such as atoms or molecules, which are made up of multiple particles with spin 1/2.

3. Can SU(2) Spinor be applied to all particles with spin 1/2?

Yes, SU(2) Spinor can be applied to all particles with spin 1/2, regardless of their physical properties. This is because it is a mathematical representation of a fundamental symmetry group, and its principles can be applied to any system with particles with spin 1/2.

4. How does SU(2) Spinor relate to the Standard Model of particle physics?

SU(2) Spinor is a key component of the Standard Model of particle physics, which is a theoretical framework that describes the behavior of subatomic particles and their interactions. In the Standard Model, particles with spin 1/2 are described using SU(2) Spinor, which allows for a more accurate prediction of their behavior.

5. Are there any practical applications of SU(2) Spinor?

Yes, SU(2) Spinor has many practical applications in the fields of quantum mechanics, particle physics, and solid-state physics. It is used to describe the behavior of subatomic particles, such as electrons, and their interactions with other particles. It also has applications in quantum computing and the study of materials with unique magnetic properties.

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