What is a Label? QM & QFT Explained

[resurgance2001]

In QM, position is an operator, while in QFT, it is a label. Could someone help me understand what is meant by the term "label" - just in the simplest terms possible? Cheers

 

[Nugatory]

In QM, position is an operator, while in QFT, it is a label. Could someone help me understand what is meant by the term "label" - just in the simplest terms possible? Cheers

It may help to know that time is a "label" in both QFT and QM. In both theories the operators correspond to observables, which are the things whose expected values we calculate from the state of the system. The labels identify a particular state of the system, the starting point for the calculation.

In QM, we can ask: What do we expect the position observable to be at a given time? What do we expect the other observables to be? This is using time as a label and treating position as an observable like the others.

In QFT, we can ask: What do we expect the value of the observables to be at a given time and at a given position? This is using time and position as labels.

 

[vanhees71]

What's meant here is that you label your degrees of freedom. E.g., if you have a spinless particle in QM, you can take the three components of the position operator as a complete set of independent observables, i.e., the Cartesian coordinates $(x_j)=(x_1,x_2,x_3)$. Here the "label" is just the index $j$ to enumerate the components. If you have a system of $N$ spinless particles, you can take the $3N$ position coordinates. Then you have a label running from $1$ to $N$. The observables are functions of time (in the Heisenberg picture also all your operators that represent observables in quantum mechanics are a function of time, and the Heisenberg picture is the most natural one in going heuristically from classical mechanics in Hamiltonian form over to quantum theory).

Now in a field theory you have a continuum theory. The dynamical variables are fields, i.e., quantities which are functions of position and time. It gives you the value of the quantity (e.g., an electromagnetic field in terms of the field-strengths components $\vec{E}$ and $\vec{B}$). This means here you have two kinds of labels, a discrete one, enumerating the components $(E_j)=(E_1,E_2,E_3)$ of the field components and the position $\vec{x}$ in space where you measure these components.

In quantum field theory thus the position arguments in the field operators are just usual number, because they are in that sense a kind of continuous label for the infinitely many degrees of freedom represented by these fields.

Note that time is always a parameter in quantum theory, no matter, whether it's a quantized point-particle system (applicable in non-relativistic quantum theory and unfortunately often called the "first quantization") or a many-body system of indefinite particle number (applicable in both non-relativistic and relativistic quantum theory), i.e., a quantum-field theory.

 

[muscaria]

In QM, position is an operator, while in QFT, it is a label. Could someone help me understand what is meant by the term "label" - just in the simplest terms possible? Cheers

This is what happens when you go from systems with a finite number of degrees of freedom to an infinite number of continuous degrees of freedom which is a field theory. This happens at the classical level before any kind of quantisation. The point is when you have a finite number of degrees of freedom, the associated coordinates are variables which encode the locations of particles and their variations represent different positions/configurations of the mechanical system in the usual sense. When you have an infinite number of continuous degrees of freedom such as a fluid, you consider your space positions to be fixed points in space which do not vary, in contrast to the previous situation where one takes the positions of the discrete system to be moving in space. In the case of a field, positions in space are fixed and do not move, instead it is the field value defined over these position labels which varies, giving rise to wave-like motion whose form depends on the kind of interactions one defines between the degrees of freedom. If you'd like an example of a field that would make this a bit more tangible, let us know!
EDIT: Only noticed vanhees71's nice post while posting!