# What is meant by a "c-number"?

In Chapter 2 of David Tong's QFT notes, he uses the term "c-number" without ever defining it.

Here is the first place.

However, it's easy to check by direct substitution that the left-hand side is simply a c-number function with the integral expression$$\Delta(x - y) = \int {{d^3p}\over{(2\pi)^3}} {1\over{2E_{\vec{p}}}}(e^{-ip \cdot (x - y)} - e^{ip \cdot (x - y)}).$$

Here is the second place, on the same page (i.e. page 37).

I should mention however that the fact that $[\phi(x), \phi(y)]$ is a c-number function, rather than an operator, is a property of free fields only.

My question is, what does c-number function mean?

• Do you want to understand c-number or c-number function? Commented Oct 11, 2015 at 8:17

A c-number basically means 'classical' number, which is basically any quantity which is not a quantum operator which acts on elements of the Hilbert space of states of a quantum system. It is meant to distinguish from q-numbers, or 'quantum' numbers, which are quantum operators. See http://wikipedia.org/wiki/C-number and the reference therein.

The term c-number is used informally in the way Meer Ashwinkumar describes. As far as I know, it doesn't have a widely promulgated formal definition. However, there is a formal definition for c-number that agrees with the way the term is used in many cases, including the case you're asking about.

As you may know, you can think of the operator formalism for quantum mechanics as a generalized version of probability theory, in which real-valued random variables are represented by self-adjoint operators on a Hilbert space. More generally, complex-valued random variables are represented by normal operators.

A c-number is a random variable represented by a scalar multiple of the identity operator.

Intuitively, a c-number is a random variable that isn't really random: its value is a constant. The identity operator itself, for instance, represents the random variable whose value is always $1$, while $-4$ times the identity represents the random variable whose value is always $-4$. You can see why this makes sense by computing the expectation value, variance, and higher moments of a c-number relative to some state.

In your example, Tong is talking about a model for a random scalar field,^ whose amplitude at the point $x$ is the real-valued random variable $\phi(x)$. For any two points $x$ and $y$, the commutator $[\phi(x), \phi(y)]$ represents an imaginary-valued random variable. The commutator turns out to be a multiple of the identity—in other words, a c-number. Since this c-number depends on $x$ and $y$, Tong calls it a c-number function (of $x$ and $y$).

^ A free scalar field can be seen as a quantum version of white noise.

• The remark "you can think of the operator formalism for quantum mechanics as a generalized version of probability theory" seems to be quite interesting. Can you please provide references to any papers describing it? Commented Mar 10, 2021 at 16:32
• @AmeyJoshi, my favorite is in Peter Whittle's book Probability via Expectation. It's the last chapter, "Quantum Mechanics." In the first few chapters, Whittle lays out the basics of probability as a theory of expectation values on commutative algebras. Once you get that, you can turn to the last chapter for the non-commutative case. Commented Mar 10, 2021 at 18:16
• Thanks a lot, @Vectornaut. The book looks quite interesting. Commented Mar 11, 2021 at 4:29

This particular "$c$-number function" is called the Pauli-Jordan Operator. You might want to peruse Ryder's Quantum Field Theory specifically §4.2 and §6.1.