# Quantum Mechanics Defining a field

I am currently in a basic class where I am being taught some concepts of QM.. We were taught today some field theory but we were not given the definition of field in QM. I looked around but my book did not state anything. Can anyone define a field in QM?

• In quantum mechanics, you usually deal with only one or a few degrees of freedom. E.g. the 1D harmonic oscillator has one degree of freedom described by a coordinate $q$. In contrast, a field has infinite degrees of freedom. Consider a field $\phi(t,x)$. At every point $x$, the field has a degree of freedom. For example if $\phi(t,x)$ is a Klein-Gordon field, you can imagine that at every point $x$, there is a quantum harmonic oscillator where now $q\rightarrow \phi(t,x)$. – honey.mustard Nov 16 '18 at 4:40
• Stay away from quantum fields if you care about your mental health! – Run like hell Nov 16 '18 at 12:44

In classical physics, a particle is a thing characterized by a single position $$\mathbf{x}$$, which may change in time: $$\mathbf{x}(t)$$.

In contrast, the simplest type of field (still in classical physics) is a function $$\phi$$ that that assigns a real number $$\phi(\mathbf{x})$$ to each point $$\mathbf{x}$$ in space, and this whole function can change in time: $$\phi(\mathbf{x},t)$$. This simplest type of field is called a (classical) scalar field. Other types of field assign something else to each point in space. For example, the electric field assigns a vector $$\mathbf{E}(\mathbf{x})$$ to each point in space, and this vector-valued function can change in time: $$\mathbf{E}(\mathbf{x},t)$$.

The preceding paragraph was all about classical physics.

In quantum physics, the distinction between particles and fields is similar, except that things generally only have definite values when we measure them.

A quantum particle is described by a set of time-dependent "operators" $$X_k(t)$$ that are analogous to the coordinates of a classical particle, except that they generally don't have definite values except when we measure them.

Similarly, a quantum field is analogous to a classical field, except that now instead of assigning a number (or a vector's worth of numbers) to each point in space and time, it assigns an operator to each point in space and time. The value of the field at a given point in space generally doesn't have a definite value except when we measure it.

For quantum particles, the uncertainty principle says that there is a limit to the degree to which the particle's position and momentum can both be well-defined at the same time. The more sharply-defined one of them is, the less sharply-defined the other one must be.

For the quantum electromagnetic field, a similar type of uncertainty principle says that there is a limit to the degree to which the electric and magnetic fields can both be well-defined at the same point in space and time. The more sharply-defined the electric field is, the less sharply-defined the magnetic field must be, and conversely.

Appendix

The preceding description used the word "operator" without explaining what those operators operate on. This appendix gives a mathematically-explicit example to try to clarify this.

In single-particle QM, the operator $$X_k$$ corresponding to the $$k$$-th coordinate of the particle's location acts on the wavefunction $$\psi(\mathbf{x})$$ like this: $$X_k\psi(\mathbf{x})=x_k\psi(\mathbf{x}), \tag{1}$$ where $$x_k$$ is the $$k$$-th component of $$\mathbf{x}$$. The momentum operator $$P_k$$, which is proportional to the time-derivative of $$X_k$$ in the Heisenberg picture, acts on the wavefunction like this: $$P_k\psi(\mathbf{x})\propto \frac{\partial}{\partial x_k}\psi(\mathbf{x}). \tag{2}$$ The "uncertainty principle" is one manifestation of the fact that the operators $$X_k$$ and $$P_k$$ do not commute with each other: $$X_k P_k\psi(\mathbf{x})\neq P_k X_k\psi(\mathbf{x}). \tag{3}$$

The simplest type of quantum field is a scalar field. To avoid technical complications, this is easiest to describe if we replace continuous-and-infinite space with a discrete-and-finite lattice. We can take the lattice to by very fine and very large so that it might as well be continuous and infinite. In this context, $$\mathbf{x}$$ can be interpreted as a collection of indices that specify a site in the lattice. The wavefunction in this case is a function of many real variables. For each lattice site $$\mathbf{x}$$, we have one real-valued variable $${r}(\mathbf{x})$$. The wavefunction is a function of all of these variables: $$\psi\big({r}(\mathbf{x}_1),{r}(\mathbf{x}_2),...\big) \tag{4}$$ where the list of arguments includes one variable $${r}(\mathbf{x}_n)$$ for each lattice site. This is often abbreviated $$\psi[{r}], \tag{5}$$ where the square brackets are meant to remind us that the argument is really a very long list of arguments, one for each lattice site. For each lattice site $$\mathbf{x}$$, we have a field operator $$\phi(\mathbf{x})$$ acts on the wavefunction like this: $$\phi(\mathbf{x})\Psi[{r}] = {r}(\mathbf{x})\psi[{r}]. \tag{6}$$ This is analogous to equation (1). In the Heisenberg picture, these operators change with time, and equation (6) represents only how they operate when $$t=0$$. The time-derivative of the field operator at $$t=0$$ acts like this: $$\dot\phi(\mathbf{x})\psi[{r}] \propto \frac{\partial}{\partial {r}(\mathbf{x})}\psi[{r}]. \tag{7}$$ This is analogous to equation (2); and just as in that analogy, the operators $$\phi(\mathbf{x})$$ and $$\dot\phi(\mathbf{x})$$ do not commute with each other.

The effect of the operators $$\phi(\mathbf{x},t)$$ when $$t\neq 0$$ is more complicated (as is the effect of $$X_k(t)$$ in single-particle QM), and I won't try to describe it explicitly here. (It involves expressing the Hamiltonian $$H$$ in terms of the field operators at time $$t=0$$ and then using that operator in the usual way to describe the effect of time-evolution.) The goal here is only to convey the basic idea of what a field operator operates on.

The quantum electromagnetic field is a little more complicated, but here's a schematic description. In $$D$$-dimensional space, for each lattice site $$\mathbf{x}$$, we have a set of $$D$$ real-valued variables $$A_k(\mathbf{x})$$. Actually, it is better to think of this as a single real variable associated with each link (pair of neighboring sites), where $$\mathbf{x}$$ is one of the sites in the pair and the index $$k$$ specifies the direction to the neighboring site. The wavefunction $$\psi$$ is a function of all of these variables, abbreviated $$\psi[A]$$. The magnetic field operator has components $$B_{jk}(\mathbf{x})$$ that act like $$B_{jk}(\mathbf{x})\psi[A]\sim \big(\nabla_j A_k(\mathbf{x}) - \nabla_k A_j(\mathbf{x})\big)\psi[A]$$ where $$\nabla_k$$ is a discrete version of the gradient with respect to $$\mathbf{x}$$, so $$B_{jk}$$ is associated with a plaquette (a little square made of four neighboring lattice sites). The electric field operator has components $$E_k(\mathbf{x})$$ that act like $$E_{k}(\mathbf{x})\psi[A]\propto\frac{\partial}{\partial A_k(\mathbf{x})} \psi[A].$$ I'd better stop here, because this post is already much longer than originally planned.

• re. "it assigns an operator to each point in space and time" what is this operator? what functions does it operate on? – user45664 Nov 16 '18 at 16:32
• @user45664 I added an appendix to help explain what those words mean. It's mathematical, and even though it's long it still omits lots of important detail (like how to handle things at different times), but hopefully it at least begins to convey the idea. – Chiral Anomaly Nov 16 '18 at 17:18

A field doesn't really exist in quantum mechanics, which deals with one or a finite number of particles. But to be brief, a field $$\phi(t,\vec{x})$$ is an Hermitian operator on Hilbert space (say Fock space) associated to a specific time $$t$$ and position $$\vec{x}$$. The $$t$$-dependence of $$\phi$$ is governed by the Hamiltonian $$H$$, i.e. $$\phi(t,\vec{x}) = e^{i H t}\phi(0,\vec{x}) e^{-i H t}$$. Likewise, shifts in $$\vec{x}$$ are governed by generators $$\vec{P}$$ of translations. If you think of the harmonic oscillator (or an interacting version of it) as a 0+1-dimensional theory, the field $$\phi(t,\vec{x})$$ is the $$d$$+1-dimensional cousin of the operator $$X = (a + a^\dagger)/\sqrt{2}$$, where $$d$$ is the dimension of space.