Secound quantization

Second quantization is just a way of reformulating Schroedinger's equation (Eq 1.1, 1.30) for many identical particles as a quantum field theory. It's just the same theory rewritten, so there is still a Hamiltonian (Eq 1.32) and states (Eq 1.33, 1.34). As he writes after Eq 1.34, "It is now straightforward (though tedious) to verify that eq. (1.1), the abstract Schroedinger equation, is obeyed if and only if the function ψ satisfies eq. (1.30)."

http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf

 

No, it's not the same theory.

Write down the Dirac-Lagrangian. In rel. QM the ψ(x) is a wave function, in QED ψ(x) becomes an operator acting on states. That's different.

In QM you have states |ψ> and wave functions ψ(x) = <x|ψ>.

In QED ψ(x) is itself an operator.

 

The place where "ψ(x) as operator" diverges from "ψ(x) as wavefunction" is the step where we reinterpret the negative frequency modes as positive energy antiparticles. Since ψ is now a hybrid quantity that creates sometimes and destroys other times, it is no longer possible to think of it simply as the probability amplitude of finding something.

 

You can construct a quantum field theory w/o referring to frequencies and w/o such an interpretation. Historically that was the way QFT was developed, but it's not a necessary ingredient for a construction.

 

There seem to be two meanings of "second quantization". The first presentation seems more common in condensed matter textbooks. The second seems more common in particle physics textbooks.

In the case of the non-relativistic Schroedinger equation for many identical particles, they are the same theory. The theory already works, but "second quantization" is a dual formulation that makes it easier to see some things.

In the case of the Dirac equation, the theory of a single particle is useful, but isn't completely sensible because of the kludge of the filled sea of antiparticles. There "second quantization" yields a different theory, of a field, which does work.

I think both procedures are called "second quantization" because one ends up with a quantum field theory. However, in the first case, the field theory can be mathematically derived from the many-body Schroedinger equation. In the second case, it is not justified (since the Dirac equation doesn't work without the kludge), and second quantization is not a "derivation", but should be seen rather as postulating a new theory.

 

Aren't these two remarks referring to the same thing? I'd say yes, you can write down QFT without the negative-frequency reinterpretation, and historically it was done that way, but it turned out to be incorrect.

Yes, Srednicki was discussing a nonrelativistic QFT, for which the antiparticle issue doesn't arise.

 

@Bill_K, yes, that's my understanding. I'm always amazed by how they ended up at the right theory by such a wrong route. I believe the condensed matter second quantization came even later, even though it is more "elementary", being "just" Schroedinger's equation.

 

What I am saying is that you can define a QFT (QED, QCD, ...) in position space w/o ever referring to momentum space or frequencies. Then you can transform a QFT (e.g. gauge fixed QED, QCD) to momentum space and introduce creation and annihilation operators (you need to cheat a little bit b/c of Haag's theorem). The structure of the theory follows mathematically w/o referring to a physical interpretation.

 

What I thought was, you can't define the vacuum state without reference to frequencies, and without knowing the vacuum state you don't have a complete QFT. When you write the expansion of ψ into normal modes you need some way of identifying which of the operators are going to be creation operators and which are annihilation operators, so you have to know which of the modes are positive frequency and which are negative. The vacuum state is then defined as the state on which the annihilation operators give 0.

This is the whole point behind Hawking radiation, for example, in which several different vacuum states can be defined, and there is a Bogoliubov transformation between them that reverses the nature of some of the operators.

 

I don't fully agree; what you are describing is the Fock vacuum |0> with

[tex]a_n|0\rangle = 0[/tex]

Yes, you need positive and negative frequencies, but this is an artefact of the definition of the theory.

The "true" vacuum state |Ω> should be defined as something like

[tex]\text{min}_\Omega\,\langle\Omega|H|\Omega\rangle [/tex]

|Ω> may not be identical to the trivial Fock vacuum |0>, in QCD they are different.

I agree that in applications like perturbation and scattering theory you may use |0> w/o problems

 

To go from the classical Hamiltonian to the correct quantum Hamiltonian, don't you need to normal order it? And to do that, you again need to split the field into positive and negative frequency parts, ψ(x) = ψ⁽⁺⁾(x) + ψ^(-)(x). For without normal ordering, the Hamiltonian will not have a minimum?

 

First quantization is quantum mechanics where the total number of particles in the system is fixed. For instance, in an atom there can be 12 electrons in orbiting a nucleus with 12 protons. The wave functions corresponding to the positions of these 12 electrons can be calculated. However, there will always be a total of 12 electrons in this calculation. Usually, the field strength of the electromagnetic field will also be set constant. This implies that the number of photons is fixed.
One could estimate the probability of an electronic transition using first quantization. However, you can not calculate the probability of a new electron forming in a collision.
Second quantization is quantum mechanics where the total number of particles in the system changes. For instance, suppose a gamma wave with 1.06 meV energy collides with that atom, which initially has 12 electrons. An electron - positron pair may be created, so that there are 13 electrons and one positron. Furthermore, the photon may disappear when the electron-positron pair are created. This will change the field strength of the electromagnetic wave. The number of electrons, the number of positrons, and the number of photons have changed.
First quantization is a special case of second quantization. First quantization is basically an approximation of second quantization, where the number of particles doesn't change.

 

Second quantization was first developed by Jordan (1927) as an operator formalism for a system of n identical particles which automatically takes into account the symmetrization/antisymmetrization. Its aim was to act as a simpler replacement for the description by Slater determinants.

As the theory of matter's interaction with the quantized field was developed in the 1930's, second quantization was generalized by Fock (1932) to systems for which the number of particles is not a constant of the motion.

 

Let me give you an example.

Maxwell’s equations describe the EM field and their solutions inform us about the form of the EM field as well as about its energy, linear and angular momentum, etc. But, Maxwell’s theory allows continuous values for the above quantities and, as we have learned from various experiments, these quantities can take values only from a discrete set, i.e. they are quantized. So, how can we “fix” Maxwell’s theory, in order to obtain the correct results? There is a technique called “quantization” and this can be achieved by many equivalent ways. One of them is the so called “canonical quantization” according to which we interpret the EM fields as operator-valued functions of spacetime and we axiomatically impose the canonical commutation relations between these operators. These operators are considered to act on a state which inform us about the energy, momentum etc. of the EM field. But, as a mathematical consequence of the canonical commutation relations, energy (and the other physical quantities) can only take discrete values and the total energy of the field is a sum of these discrete values. These energy quanta are then interpreted as particles (photons) and the EM field is considered to be consisted from a finite collection of these particles.

Now, what about the rest of the observed particles? We could apply the above prescription, but we have not some field equations that describe these entities. And this is exactly what QM provide us: some field equations that describe the entity we want to study (KG equation, Dirac equation, etc.) . But these equations do not inform us about the totality of these entities or about the ways that they can be created or annihilated. So what do we do? Now that we have the field equations that describe an entity, we just quantize the corresponding fields, by applying the previous prescription (canonical quantization). That’s the meaning of interpreting a wave function as an operator-valued function of spacetime. By doing this, we eventually find that the energy of the field we have quantized, is a sum of a finite collection of energy quanta, which are then interpreted as particles.

P.S. This technique is called “second quantization” because we apply once the rules of QM, (promoting the dynamic variables to operators) in order to get the desired field equations and then we apply again the same rules to the fields (which now are interpreted as operators) in order to explain the presence of the indivisible quanta (particles).

 

I have to repeat what I've already said above several times. The field operator is not just an operator-valued interpretation of the original wavefunction. That's the way things started out, but it turned out to be wrong.

ψ is a hybrid quantity, related to not just one type of particle but two. The solutions of relativistic wave equations have a fatal flaw, namely negative energy modes. You can't just discard them, because that way leads to faster-than-light propagation. And you can't just go ahead and quantize the equation as it stands, because that fails to cure the problem. The remedy is to invent a field operator ψ that creates one type of particle and destroys another. Having done that, the operator ψ no longer bears a simple, obvious connection to the original unquantized field ψ(x). It's a new concept.