Different quantum fields for different particles?

Question

I started my studies about quantum field theory but I have some problem to understand the whole concept. I have the following questions:

1. The Lagrangian contains a dirac term for fermions and a maxwell term for gauge bosons. But does this mean that there is only one dirac field for all fermions? I guess this is not the case but I don't understand why we do not sum over the masses and other quantum numbers in the Lagrangian to distinguish the fields?

2. We interpret particles as excitations of the field. Should we assume that there is one "field" for each particle or do we create for example three electrons into a common field? But what about the pauli exclusion principle in this case? Is it impossible to have two fermions of the same kind (and same quantum numbers) in the whole universe (the occupation number of fermions is 0 or 1)?

Answer 1

1. In case you have different types of elementary particles, you must add different fields in the Lagrangian. You will have multiple Dirac terms like this: $$ \mathcal{L} = \bar{\psi}_1 (i \gamma^{\mu} \partial_{\mu} - m_1) \psi_1 + \bar{\psi}_2 (i \gamma^{\mu} \partial_{\mu} - m_2) \psi_2 + \dots. $$

2. This only is true for different types of elementary particles (e.g. electron, muon, tau). But the theory of the quantum electron field is capable of describing multiple electrons on its own, no extra fields are needed.

3. Look up "Fock space". For bosons, it consists of symmetrized tensor products of 1-particle states: $$ \mathcal{H} = \mathcal{H}_0 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \dots, $$ where $$ \mathcal{H}_n = \text{Sym}\,\left[ \mathcal{H}_1 \otimes \dots \otimes \mathcal{H}_1 \right].$$ Let me explain the notation. $\mathcal{H}_1$ stands for the 1-particle Hilbert space (space of particle wavefunctions, etc), whereas $\text{Sym}$ is symmetrized tensor product. Symmetrization is a direct consequence of particle interchangeability: there's no "first" and "second" particle, both are equivalent.

4. For fermions (e.g. electrons) we use antisymmetrization instead: $$ \mathcal{H}_n = \text{Assym} \, \left[ \mathcal{H}_1 \otimes \dots \otimes \mathcal{H}_1 \right].$$ Interchange between two fermions changes the sign of the wavefunction. This is, of course, unobservable directly (only projective classes of wavefunctions matter in QM), so fermions are also interchangeable. But it reflects the Fermi-Dirac statistics.

5. Note also that the Pauli principle follows directly from fermionic Fock space. E.g. for two fermions, $$\mathcal{H}_2 = \text{Assym}\, \left[ \mathcal{H}_1 \otimes \mathcal{H}_1 \right]. $$ Its elements are wavefunctions $$ \Psi(x_1, x_2) $$ of two arguments (generalized arguments denote both continuous momenta quantum numbers and discrete internal spin quantum numbers) satisfying $$ \Psi(x_1, x_2) = - \Psi(x_2, x_1). $$ This means that $$ \Psi(x_1, x_2) = 0 \quad \text{when} \quad x_1 = x_2, $$ which is precisely the exclusion principle in the mathematical form.