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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?

Anon21
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  • Your definition is correct, but your conclusion isnt. Why do you think the exp function maps imaginary numbers to imaginary numbers? – lalala Jul 27 '19 at 16:00
  • Consider to only ask 1 subquestion per post to keep the discussion simple. – Qmechanic Jul 27 '19 at 16:57
  • Was the motivating assumption behind this question that $e^{ix}$ is a purely imaginary value? – Nat Jul 27 '19 at 17:15

5 Answers5

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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.

G. Smith
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    Both definitions are perfectly fine, and they are equivalent (under $J\to i J$). The current $J$ is an external parameter, we do not integrate over it, and we are free to redefine it at will. It need not be real. And, strictly speaking, it need not be complex either: one may regard it as a formal parameter with respect to which one can take derivatives; the whole purpose of $Z[J]$ is that the n-point function is $Z^{(n)}[0]$ (and, in fact, from this POV the first def. is the best one; with the second one the correlator is $(-i)^nZ^{(n)}[0]$). – AccidentalFourierTransform Jul 27 '19 at 22:55
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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).

MadMax
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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.

Matthew Roh
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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.

sweetser
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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

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

Anon21
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