Why can't quantum field theory be complex instead of imaginary?

Question

In the following question 1, the author claims that a QFT is defined as:

$$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$

Then uses this definition to explore the possibility of formulating a QFT using the quaternions, on the grounds that it is constructed over the complex numbers thus why not try to extend it.

Is this definition of $Z$ correct? My understanding is that the correct definition is:

$$Z[J] \propto \int e^{i \left( S[\phi]+J.\phi \right)} D[\phi]$$

In this case, the exponentiated term $i(S[\phi]+J.\phi)$ is not a complex number, but only an imaginary part.

The original author asks: "Why can't quantum field theory be quaternion instead of complex?"

First, I would like to confirm if the author's definition is or isn't an error. Then, assuming that is it an error, I would like to ask the intermediary question: is there any possibility of a QFT which admit a real scalar within the exponential term in addition to the imaginary term, such that the sum is over the complex numbers and not just the imaginary part?

Answer 1

Quantum field theory is complex, not purely imaginary. If the action $S$ is real, then $e^{iS}$ is a unit-magnitude complex number lying on the unit circle in the complex plane.

By Euler’s formula,

$$e^{iS}=\cos{S}+i\sin{S}.$$

Your second definition of $Z$ is the correct one.

Answer 2

Apart from the caveats pointed out by the other answers, the underlying assumption that the action $S$ is real is not necessarily true.

For example, the kinetic portion of a Dirac spinor $$ S = \int i\bar{\psi}\gamma^\mu\partial_\mu\psi $$ is purely imaginary (multiplication of Grassmann pairs).

Answer 3

It is complex. If you thought $e^{ix}$ was purely imaginary, look at $e^i$ on Wolfram|Alpha. It isn't purely imaginary, it has its real parts too. The values of $x$ when $e^{ix}$ is purely imaginary, is very limited, in fact such values of $x$ are when it is a very distinct multiple of $\pi$, for example, $\pi$, $2\pi$, $0.5\pi$, and so on.

Answer 4

There are different ways to represent complex numbers and quaternions. The exponential with the factor of $i$ for complex numbers and $(i, j, k)$ for quaternions represents the same same set of numbers as the more familiar representations with two or four numbers. This is Euler's formula put to use, $e^{i \alpha}=\cos(\alpha)+i \sin(\alpha)$. The only complication in using this directly with quaternions is that $i$ goes to $a i + b j + c k$ where these three together have to have a norm of one.

Quaternions cannot be used for quantum mechanics because one cannot form spin states with the division algebra where two non-zero states are orthagonal. One can do some work with quaternion series. This math structure is made up of n total quaternions which one then says has u columns and v rows such that u*v=n. A quaternion series is not a division algebra but is a semi-group with inverses. Two quaternion series can be non-zero and orthogonal Most of the basic properties of a spin state are easy to show with quaternion series. The inner product of a bra 1x3 and a ket 3x1 results in a scalar, a 1x1 quaternion series. The scalar can be complex-valued or quaternion-valued. The craft of quantum mechanics is to find operators on states such that the scalar is real-valued. All tools of quantum mechanics necessarily can be reconstructed with quaternion series so long as each state has the form (,,0,0). Complex numbers are a subgroup of quaternions. Quantum mechanics when viewed this way appears odd - why don't we point in a different direction? The reason is that quantum mechanics studies a very particular tiny volume of space-time and that tiny volume is in one particular direction in space-time. Quaternions that all point in the very same direction all commute.

Answer 5

A QFT can be constructed as

$$Z[J] \propto \int e^{iS[\phi]+J.\phi} D[\phi]$$

because

$$ e^{iS+J} = e^Je^{iS}=e^J(\cos S + i \sin S) $$

The real part of the exponential simply represents the ground-state/degeneracy and is absorbed in the normalization constant.

The most general definition would then be

$$Z[J] \propto \int e^{i(S[\phi]+J.\phi)+V[\phi]} D[\phi]$$