Dot product of two pauli matrices

In summary, in a linear vector space, matrices can be a special kind of "vectors" as long as they follow the rules of linear space. In quantum mechanics, authors often use "vectors" with three pauli matrices Sx, Sy, Sz. The dot product between these "vectors" can be defined using the anticommutator or by coupling it with a trace operation to produce a true scalar instead of a scalar times identity. The cross product or wedge product can also be defined using the commutator.
  • #1
KFC
488
4
In some text, I read something like this

[tex]\vec{S}_i\cdot\vec{S}_j[/tex]

where [tex]\vec{S}_i[/tex] and [tex]\vec{S}_j[/tex] are "vectors" with each components be the pauli matrices [tex]S_x, S_y, S_z[/tex] individularly. My question is: if all components of this kind of vector are a 3x3 matrix, so how do you carry out the dot product? So the dot product will give another matrix instead of a number?
 
Physics news on Phys.org
  • #2
"Vectors" are not matrices. Matrices are tensors of rank 2, while vectors are tensors of rank 1. I don't see how you can have "vectors" which are Pauli-Matrices.

Matrix multiplication (2 matrices multiplied together, not a matrix and a vector) has specific rules and will result in another matrix with the same number of columns and rows.
 
  • #3
Matterwave said:
"Vectors" are not matrices. Matrices are tensors of rank 2, while vectors are tensors of rank 1. I don't see how you can have "vectors" which are Pauli-Matrices.

Matrix multiplication (2 matrices multiplied together, not a matrix and a vector) has specific rules and will result in another matrix with the same number of columns and rows.

In the text of linear space, it read: matrices can be a special kind of "vectors" (that's why we add qutoation mark). In abstract linear vector space, anything satisfying the RULES of linear space can form a vector. And in many QM text, the author like to form a "vector" with three pauli matrices Sx, Sy, Sz. What I am really confuse is how to carry out the dot product b/w such "vectors"?
 
  • #4
Hmmm, I don't know if the dot product is the same as matrix multiplication (I'm not sure how the inner product is defined in a vector space that includes matrices), but if it is then:

http://en.wikipedia.org/wiki/Matrix_multiplication

That explains it pretty well.

Oh, I was thinking of only square matrices before, if the matrix is not square then the resulting matrix won't have the same dimensions as before. Sorry, it's been a while since I've really had to do Matrix multiplication. :P
 
  • #5
A logical way to define a dot product when using pauli matrixes as basis vectors would be to use the anticommutator

[tex]
a \cdot b = \frac{1}{2} \{ a, b \} = \frac{1}{2} (a b + b a )
[/tex]

EDIT: latex in PF doesn't appear to be working right now. That was:
a \cdot b = \frac{1}{2} \{ a, b \} = \frac{1}{2} (a b + b a )Such a dot product won't be in the span of the pauli matrixes themselves, but will be your typical vector dot product multiplied by the identity matrix. If you wanted a dot product that produced a true scalar instead of scalar times identity, then you should be able to couple the above with a trace operation:

[tex]
a \cdot b = \frac{1}{4} TR \{ a, b \} = \frac{1}{4} TR (a b + b a )
[/tex]

EDIT: latex in PF doesn't appear to be working right now. That was:
a \cdot b = \frac{1}{4} TR \{ a, b \} = \frac{1}{4} TR (a b + b a )

You can similarily and naturally define the cross product or wedge product using the commutator. Some playing with these ideas can be found here:

http://sites.google.com/site/peeterjoot/math/pauli_matrix.pdf

( as written this assumes some clifford algebra background ).
 
  • #6
Thanks a lot. It helps.
Peeter said:
A logical way to define a dot product when using pauli matrixes as basis vectors would be to use the anticommutator

[tex]
a \cdot b = \frac{1}{2} \{ a, b \} = \frac{1}{2} (a b + b a )
[/tex]

EDIT: latex in PF doesn't appear to be working right now. That was:
a \cdot b = \frac{1}{2} \{ a, b \} = \frac{1}{2} (a b + b a )


Such a dot product won't be in the span of the pauli matrixes themselves, but will be your typical vector dot product multiplied by the identity matrix. If you wanted a dot product that produced a true scalar instead of scalar times identity, then you should be able to couple the above with a trace operation:

[tex]
a \cdot b = \frac{1}{4} TR \{ a, b \} = \frac{1}{4} TR (a b + b a )
[/tex]

EDIT: latex in PF doesn't appear to be working right now. That was:
a \cdot b = \frac{1}{4} TR \{ a, b \} = \frac{1}{4} TR (a b + b a )

You can similarily and naturally define the cross product or wedge product using the commutator. Some playing with these ideas can be found here:

http://sites.google.com/site/peeterjoot/math/pauli_matrix.pdf

( as written this assumes some clifford algebra background ).
 

1. What is the dot product of two Pauli matrices?

The dot product of two Pauli matrices is a mathematical operation that results in a scalar quantity. It is calculated by multiplying the corresponding elements of the two matrices and summing them up.

2. Why is the dot product of two Pauli matrices important in quantum mechanics?

The dot product of two Pauli matrices is important in quantum mechanics because it is a fundamental operation used in the calculation of various quantum mechanical quantities, such as expectation values and transition probabilities.

3. How is the dot product of two Pauli matrices related to spin in quantum mechanics?

In quantum mechanics, the dot product of two Pauli matrices is related to spin as it represents the spin operator in the x, y, or z direction. The result of the dot product is used to calculate the spin of a particle in a particular direction.

4. Can the dot product of two Pauli matrices be used to calculate entanglement?

Yes, the dot product of two Pauli matrices can be used to calculate entanglement between two quantum systems. The result of the dot product is related to the correlation between the two systems, which is a measure of their entanglement.

5. Are there any properties of the dot product of two Pauli matrices that are useful in quantum mechanics?

Yes, the dot product of two Pauli matrices has several properties that are useful in quantum mechanics, such as being Hermitian and unitary. These properties make it a useful tool for calculating and analyzing quantum systems.

Similar threads

Replies
14
Views
1K
Replies
5
Views
3K
Replies
9
Views
1K
  • Quantum Physics
Replies
17
Views
2K
Replies
4
Views
3K
  • Quantum Physics
Replies
1
Views
2K
  • Quantum Physics
Replies
1
Views
774
Replies
14
Views
1K
  • Quantum Physics
Replies
2
Views
1K
Back
Top