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I have been recently learning second quantisation which makes me wonder what is actually 'quantised' in first quantisation. I have read many posts where it is stated that representing classical phase space variables by operators on a Hilbert space and thus solving the Schrodinger equation is the essence of first quantisation (correct me if I am wrong). However, it still appears to me somewhat vague that 'quantisation' of which quantity does this term refer to? Why is the position basis representation of a quantum many body state or an operator called its 'first quantised' form?

I understand that by doing the above, we get the stationary states of the quantum Hamiltonian with a discrete or continuous spectra of energy. In simpler words, energy is quantised: only certain values of energy are allowed for a particular quantum system. The photoelectric effect is a direct proof of light being consistent of photons having quantised energy. I have studied basic quantum mechanics and know that the angular momentum of an electron in a Hydrogen atom has certain values allowed by the system. Hence angular momentum is also quantised. But all these ideas of quantisation refer to different physical quantities. Do these altogether comprise the idea of first quantisation or is there any one specific physical or mathematical definition of first quantisation?

I am at a very preliminary stage of learning quantum field theory. As of yet, I have read that second quantisation means introducing the Fock basis states, creation-annihilation operators and their commutation relations and then writing all operators and states in their terms. I also know, that these operators and states can also be expressed in terms of fields.

I have looked into Is a "third quantisation" possible? and First quantisation vs Second quantisation. However, I am looking for an easier explanation of precisely first quantisation, if possible remaining restricted to basic quantum and classical mechanics.

Alice
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The terms first and second quantization are IMHO pretty misleading.

In this context, quantized does not mean "turning a continuous physical variable into a discrete one", but "transforming a classical system into a quantum one" : quantization refers to a procedure to transform a classical system into a quantum counterpart. If the classical system is described by a point in phase space $M$, the quantum system is described by a wave function, which is a square-integrable function from $M$ to $\mathbb C$.

In a quantum system, some physical quantities are discrete while others are not.

Historically, first quantization refers to this procedure being applied to finite dimensional phase spaces (like some particles moving in euclidean space) and some robust mathematical tools where develop to do just this.

Second quantization originally refers to this being applied to fields like Klein-Gordon, Dirac or electro-magnetic fields. Since the Klein-Gordon and Dirac equations were attempts at a relativistic Schrödinger equation, this can give the impression that you are quantizing the wave function of a quantum system, but this is not the case.

Now, the term second quantization also appears when discussing constructing a Fock space from a $1$-particle Hilbert space. In this case, you are usually starting with a Hilbert space which is the quantization of a classical particle, and building a larger Hilbert space (called the Fock space) which is also the quantization of a classical field.

SolubleFish
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In the first quantization particles are being assigned wave-like properties. The rest is math. Of course, assigning wave-like properties does not make sense for, e.g., the electromagnetic field, since it is already waves in classical physics. In the second quantization waves are becoming particle-like. Thus, it is second only for the particles that underwent the first quantization, but actually first for the em field. Still, the term second quantization also designates a well-defined mathematical procedure, which is the same for the waves described by Maxwell and Schrodinger esuations, so it is better not to insist on its literal meaning.

Roger V.
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