WTF, QFT?

The following equation can be viewed in a couple of ways
$$
action =  \int_{\Omega} \mathcal{L} (\phi_a, \partial_{\mu}{\phi_a}) d^4x
$$

1. Classically

Every physical law can be written in terms of an Action Principle.  An action principle states that measurable values $\phi_a(\mathbb{x},t)$ over a region of spacetime $\Omega$ will be such that the action is stationary.  Or to put it another way, if you infinitesimally deform the $\phi_a$ then either

1.  however you do it the action will increase, or
2.  however you do it the action will decrease.

There is an important caveat though: the action is stationary because we only consider deformations of the $\phi_a$ that preserve its values on the boundary $\partial \Omega$.

This is an alternative to the differential way of describing the universe, in which only a single point of spacetime is considered.  In the differential formulation, instead of being told the values of $\phi_a$ on a boundary of a region of spacetime, and asked to calculate the values on the interior, we are told the values of $\phi_a$ and $\partial_{\mu} \phi_a$ at a single point, and asked to calculate their values everywhere else.  The tricky bit, if you are trying to reformulate laws as an action principle, is guessing the Lagrangian density $\mathcal{L}$.

For a concrete example, consider the following Lagrangian density
$$
\begin{align}
\mathcal{L} &= \mathcal{L}(E^i,B^i) \\
&= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}
\end{align}
$$
where $F$ is the electromagnetic tensor derived from the electric and magnetic fields $E$ and $B$.

It may not be obvious, but the action principle that $\int_{\Omega} \mathcal{L} d^4x$ is stationary is mathematically equivalent to Maxwell's differential laws for electomagnetic fields in empty space.

2. Quantum mechanically

The key to understanding the equation quantum mechanically it to figure out what kind of mathematical object $\phi_a(\mathbb{x},t)$ is.   In the classical version we know what it is: it is a number representing a measurable, for example the strength of the electric field at $(\mathbb{x},t)$.  It's a measurable in the sense that you can perform an experiment to measure it.  For example, to measure the strength of an electric field you could rub a balloon to give it some charge, attach it to a spring, hold on to one end of the spring and measure how much it is stretched.

The key idea is that if we know the state - everything there is to know about the system - then these classical numbers (or c-numbers as Dirac called them) can be predicted.  However, quantum mechanics tells us that sometimes you can know the whole state but be unable to predict the results of measurements.  For example, if you know the spin is up then you know the complete state of a single electron system but you can't predict the outcomes of spin measurements along other axes.  All you can do is to predict the probabilities for each possible outcome.  Dirac tried to capture this concept in mathematical objects he dubbed q-numbers.  In this formulation the state is assumed to be a normalized vector in a type of vector space called a Hilbert space.  And the q-numbers are special operators$^{\dagger}$ which map states to states.  Like c-numbers, q-numbers can be added and subtracted and multiplied.  But how exactly do they capture the nature of measurement in quantum mechanics?

Each operator has a set of linearly independent eigenvectors which spans the Hilbert space and has real eigenvalues $^{\dagger_2}$.  The eigenvalues are the possible results of measurements - so immediately we see that q-numbers capture the idea of quantization.  But how is the idea of uncertainty captured?  Well, the initial state is necessarily a linear combination of the eigenvectors and the coefficients are the probability amplitudes for the measurement returning the corresponding eigenvalues.  This means that if the original state is itself an eigenvector then the result of the measurement is guaranteed.  (So, for example, measuring spin along the z-axis will definitely produce a +1 if the spin starts off in state up.)  Finally, q-numbers capture the idea that making the observation necessarily disrupts the state in that after the measurement the state is changed to the eigenvector corresponding to the eigenvalue you obtained by making the measurement.

The Schrodinger equation describes how the state $\lvert \psi \rangle$ of a system changes with time, in terms of a special q-number H which measures the total energy of the system:
$$
i\hbar \frac{d \lvert \psi \rangle}{dt} = H \lvert \psi \rangle
$$
So, for example, suppose $\lvert \psi(0) \rangle$ is an eigenvector of the position q-number $X$, then you can use the equation above to determine $\lvert \psi(t) \rangle$ at some later time.  But $\lvert \psi(t) \rangle$ can be written as a linear combination of the eigenvectors of $X$ - i.e. as a linear combination of the states $\lvert x \rangle$ where position is known - so we can write
$$
\lvert \psi(t) \rangle = \int \psi_t(x)\lvert x \rangle dx
$$
And if you plot the $\psi_t(x)$ you typically get pictures showing that the electron becomes more and more delocalized as time progresses.

The above is very useful in many cases, but eventually hits a wall because:

1. It's hard to make the theory relativistic since space and time are on a different footing
2. It's hard to describe particle creation and destruction since the existence of a fixed number of particles is assumed by the operators (like X) which are defined at the outset.

Part of the problem is that we have defined the state to be time dependent, but not position dependent: $\lvert \psi(t) \rangle$ describes the entire state at time t.  Another part of the problem is that there's an operator X but no operator T.  To begin to solve this we first ditch the Schrodinger view (and the Schrodinger equation) and adopt the Heisenberg view.  In the Heisenberg view the state is fixed for all time (say $\lvert \psi_0\rangle$), but the operators are time dependent.  So if we have a q-number A then $A_{Heis}(t)$ is defined in such a way that the expected value of the measurement at time t $\langle \psi(t) \lvert A_{Schr}\lvert \psi(t) \rangle$ is the same as $\langle \psi_0 \lvert A_{Heis}(t)\lvert \psi_0 \rangle$.  The next step is to replace the operator $X(t)$ - which is a q-number telling us where a given electron is - with a matter field operator $\phi(x,t)$ - which is a q-number telling us how many electrons are in a given location.  This neatly solves both 1 and 2 above.

We are finally in a position to describe in words what kind of thing $\phi_a(\mathbb{x},t)$ actually is.  It is an operator which would tell us how much matter of a particular kind (e.g. electron) there is at a given spacetime location, were we to actually measure it.  Now lets return to the original equation and try to interpret it quantum mechanically:
$$
action =  \int_{\Omega} \mathcal{L} (\phi_a, \partial_{\mu}{\phi_a}) d^4x
$$
The action principle is the principle that the values $\phi_a(\mathbb{x},t)$ will be such that the action is stationary.  But note that the $\phi_a(\mathbb{x},t)$ are operators and so the action is also an operator.  And when we say stationary we mean: for variations which do not change the values on $\partial \Omega$.

What does it mean to define the values of $\phi_a(\mathbb{x},t)$ on $\partial \Omega$?  Well let's imagine $\Omega$ to be a block of spacetime in a 2D universe representing the existence of a solid square over a fixed span of time.  Then $\Omega$ can be represented by a cube and $\partial \Omega$ is represented by the floor, ceiling, and walls.  The floor represents the contents of the interior at the start, the ceiling represents the contents at the end, and the walls represent every interaction the interior has with the rest of the universe from the start to the end.  If we say we know $\phi_a(\mathbb{x},t)$ on this surface then we are saying we know the contents of the space at the start, the contents at the end, and what went in and what went out in between.  So the action principle says that subject to the constraint that the contents are not changed at the start or end and what comes in or goes out does not change, either all smooth infinitesimal changes to $\phi_a(\mathbb{x},t)$ on the interior increase the action, or all smooth infinitesimal changes to $\phi_a(\mathbb{x},t)$ on the interior decrease the action.  But $\phi_a(\mathbb{x},t)$ is just an operator defining what might happen were you to step inside $\Omega$ and make a measurement given an initial state $\lvert \psi_0 \rangle$.  But you cannot step inside and make a measurement without changing either the state or the operators on the boundary any more than you can walk through walls.

So the quantum mechanical interpretation appears to be built on top of a counterfactual, in a way that the classical interpretation does not.  True, the classical interpretation says what you would get were you to step inside and measure (which you cannot without changing the values on $\partial \Omega$).  However, the classical interpretation claims that the measured value exists whether you measure it or not, whereas the quantum mechanical interpretation states that it doesn't exist until the measurement is made!

WTF, Perhaps the conclusion is that the operators $\phi_a(\mathbb{x},t)$ are real, but the measurements they describe are not!  Agh. 

• $^{\dagger}$ I.e. Hermitian operators
• $^{\dagger_2}$ This is one possible definition of Hermitian