Imaginary eigenvalues of gamma matrices

[center o bass]

Hi! I'm reading David Tong's notes on QFT and I'm now reading on the chapter on the dirac equation

http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf

and I stumbled across a statement where he claims that

$$(\gamma^0)^2 = 1 \ \ \Rightarrow \text{real eigenvalues}$$

while

$$(\gamma^i)^2 = -1 \ \ \Rightarrow \text{imaginary eigenvalues}.$$

I'm a bit rusty on my linear algebra and just wondered why this is necessarily true. Why does the square of a matrix being positive and negative respectively mean real and imaginary eigenvalues?

 

[The_Duck]

Suppose x is an eigenvector of the matrix A with eigenvalue a, so that Ax = ax. Then A^2 x = a^2 x.

Suppose A^2 x = x for all x (including the eigenvectors of A). Then in view of the above, all eigenvalues of A satisfy a^2 = 1. So the possible eigenvalues are +1 and -1.

Now suppose A^2 x = -x for all x. Then all eigenvalues of A satisfy a^2 = -1. So the possible eigenvalues are +i and -i.

 

[center o bass]

Ah thanks a lot! :)

 

[center o bass]

Suppose x is an eigenvector of the matrix A with eigenvalue a, so that Ax = ax. Then A^2 x = a^2 x.

Suppose A^2 x = x for all x (including the eigenvectors of A). Then in view of the above, all eigenvalues of A satisfy a^2 = 1. So the possible eigenvalues are +1 and -1.

Now suppose A^2 x = -x for all x. Then all eigenvalues of A satisfy a^2 = -1. So the possible eigenvalues are +i and -i.

Maybe you could help me with the next thing I'm stuck on? :)

After obtaining that

$$(S^{\mu \nu})^\dagger = \frac{1}4 [\gamma^\mu, \gamma^\nu]^\dagger = - \gamma^0 S^{\mu \nu} \gamma^0$$

D. Tong claims that

$$\exp ( \frac{1}{2} \Omega_{\mu \nu} (S^{\mu \nu})^\dagger ) = \gamma^0 \exp( -\frac{1}{2} \Omega_{\mu \nu} S^{\mu \nu} ) \gamma^0.$$

I tried obtaining this by doing an exponential expansion and factorizing out the gammas, plus the property that
$$(\gamma^0)^2 = 1$$

but I get problems with the even terms in the expansion.

 

[The_Duck]

I tried obtaining this by doing an exponential expansion and factorizing out the gammas, plus the property that
$$(\gamma^0)^2 = 1$$

but I get problems with the even terms in the expansion.

What problems? Could you write out what you think the expansions of the left and right-hand sides of the equations are?

 

[center o bass]

What problems? Could you write out what you think the expansions of the left and right-hand sides of the equations are?

Ofcourse.

$$\exp(\gamma^0 (-\frac{1}2 \Omega_{\mu \nu} S^{\mu \nu}) \gamma^0) = 1 + \gamma^0 A \gamma^0 + \frac{1}{2!} (\gamma^0)^2 A^2 (\gamma^0)^2 + \frac{1}{3!}(\gamma^0)^3 A^3 (\gamma^0)^3 + \ldots = \gamma^0 ( 1 + A + \frac{1}{2!} \gamma^0 A \gamma^0 + \frac{1}{3!} A^3 + \ldots )\gamma^0$$

where i have defined

$$A = -\frac{1}2 \Omega_{\mu \nu} S^{\mu \nu}$$

 

[The_Duck]

OK, two things:

1) Your expansion of the LHS isn't right because gamma-0 and A don't necessarily commute. If A and B are not-necessarily commuting matrices or operators, then

exp(AB) = 1 + AB + (1/2!)ABAB + (1/3!)ABABAB + ...

(recall that, for instance, (AB)^2 = (AB)*(AB)). But you seem to be assuming the incorrect form

exp(AB) = 1 + AB + (1/2!)AABB + (1/3!)AAABBB + ...

which is only equivalent to the correct form if A and B commute (because (AB)(AB) != AABB unless A and B commute). (Of course, you need to apply the equivalent statement for the exponential of the product of three matrices, not just two).


2) Why are there some gamma-0's inside the parentheses in your expansion of the RHS?

 

[center o bass]

OK, two things:

1) Your expansion of the LHS isn't right because gamma-0 and A don't necessarily commute. If A and B are not-necessarily commuting matrices or operators, then

exp(AB) = 1 + AB + (1/2!)ABAB + (1/3!)ABABAB + ...

(recall that, for instance, (AB)^2 = (AB)*(AB)). But you seem to be assuming the incorrect form

exp(AB) = 1 + AB + (1/2!)AABB + (1/3!)AAABBB + ...

which is only equivalent to the correct form if A and B commute (because (AB)(AB) != AABB unless A and B commute). (Of course, you need to apply the equivalent statement for the exponential of the product of three matrices, not just two).


2) Why are there some gamma-0's inside the parentheses in your expansion of the RHS?

Ah, thank you again. The problem lied in the wrong expansion :) The gamma-0's were there due to this wrong expansion. When i tried to factor out gamma-0's to the left and right of the sum gamma-0's remained at the even terms.