QFT: Exploring the Meaning of ##\phi## and ##\psi##

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In summary: The classical field \phi(x) is analogous to the wavefunction of a quantum particle. However, in QFT, the true wavefunctions are \psi's and are not functions of spacetime coordinates anymore. They are functions of creation and annihilation operators. So, the classical fields are just the building blocks of QFT and not the actual physical quantities.In summary, the conversation discusses the concept of field operators and their relationship to classical fields in quantum field theory. The second quantization formalism introduces the hermitian operator \hat{\phi}(x) and the configuration space representation \psi(x_1,x_2,...x_n) for an n-particle Fock state. The classical field \phi(x) is not necessarily her
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Hello! So I understand that in QFT and based on the second quantization, one introduce the hermitian operator ##\hat{\phi}(x)##. So, if we have a state with n particles ##|n>## we can get the configuration space representation as: ##\psi(x_1,..,x_n,t)=<0|\hat{\phi}(x_1)...\hat{\phi}(x_n)|n>## (please let me know if anything I said is wrong). Then I read that writing the action for a field, in the end we can derive ##H[\pi,\phi]##, with ##\pi## the conjugate momentum of ##\psi## and here is where I get confused. What is the meaning of ##\phi## without a hat? It means it is just a scalar field, not a field operator? So it is just as ##\phi## was previously, in the first quantization QM? And if in the end we go back to a scalar field and not an operator, what is the point of the second quantization? I.e., if we solve the Klein-Gordon equation for ##\phi(x)## we still get a plane wave, so where is the operator-like behavior of ##\phi##? And I see that both ##\phi## and ##\psi## appear in this formalism, so they can't represent both the same thing. Thank you!
 
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Firstly, a field operator [itex]\hat{\phi}(x)[/itex] is not necessarily hermitian. For example, a complex scalar field is not hermitian but the Hamiltonian is always. A more clearer statement would be [itex]\langle 0|\hat{\phi}(x_1)\hat{\phi}(x_2)...\hat{\phi}(x_n)|n\rangle[/itex] is the configuration space wavefunction [itex]\psi(x_1,x_2,...x_n)[/itex], corresponding to the n-particle Fock state.

You're mixing up quantum states with fields. For a classical field [itex]\phi(x)[/itex], it has a classical Hamiltonian (density) which is a function of the field [itex]\phi(x)[/itex], and the corresponding conjugate momentum [itex]\pi(x)[/itex] i.e., [itex]H=H(\phi,\pi)[/itex]. There is nothing called conjugate momentum of [itex]\psi[/itex]. I think, it's a typo in your question.

[itex]\phi(x)[/itex] is the classical field, a function of spacetime coordinates [itex]x[/itex]. In the quantized version, the scalar field [itex]\hat{\phi}(x)[/itex] is a promoted to the status of an operator (not necessarily hermitian). The quantized field Hamiltonian is however a hermitian operator and is given by [itex]\hat{H}=\hat{H}(\hat{\phi},\hat{\pi})[/itex].

Second quantization is a bad name. Historically, the field [itex]\phi(x)[/itex] used to be regarded as the wavefunction and Klein-Gordon equation as a relativistic generalization of Schrodinger equation. This used to be called first quantization. However, relativistic quantum mechanics has severe pathologies and therefore was later superseded by quantum field theory (QFT). In QFT, [itex]\phi(x)[/itex] is no longer the wavefunction of a relativistic particle. The wavefunctions are what you mentioned i.e., [itex]\psi[/itex].

Yes. The solution of Klein-Gordon equation can be decomposed in terms of plane-waves with amplitudes. Those amplitudes are numbers in classical field theory and promoted to creation and annihilation operators in QFT.
 
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1. What is the difference between ##\phi## and ##\psi## in QFT?

In quantum field theory (QFT), ##\phi## and ##\psi## are both fields that describe the behavior of particles. However, ##\phi## is a scalar field that represents spin-0 particles, while ##\psi## is a spinor field that represents spin-1/2 particles. In other words, ##\phi## describes particles with integer spin (such as the Higgs boson), while ##\psi## describes particles with half-integer spin (such as electrons).

2. How do ##\phi## and ##\psi## interact with each other in QFT?

In QFT, ##\phi## and ##\psi## fields can interact with each other through a process called particle scattering. This occurs when particles exchange energy and momentum with each other, causing them to change direction or transform into different types of particles. These interactions are described by mathematical equations known as Feynman diagrams.

3. Can ##\phi## and ##\psi## fields exist at the same time and place?

Yes, both ##\phi## and ##\psi## fields can exist simultaneously at the same location in space. This is because in QFT, particles are described as excitations of their respective fields, rather than as discrete objects with definite locations. So while ##\phi## and ##\psi## fields can overlap, the particles they represent do not actually occupy the same space.

4. What is the significance of the vacuum state in QFT?

In QFT, the vacuum state is the lowest energy state of a system. This state is often referred to as the "ground state" and is the starting point for calculating the energies of excited states. In the context of ##\phi## and ##\psi## fields, the vacuum state is important because it represents the absence of particles and is used as a reference point for calculating particle interactions and energy levels.

5. How does QFT relate to other areas of physics?

QFT is a framework that combines elements of quantum mechanics and special relativity to describe the behavior of particles at the subatomic level. It is used extensively in particle physics, but also has applications in other fields such as condensed matter physics, astrophysics, and cosmology. QFT has also been instrumental in developing our understanding of fundamental forces, such as the strong and weak nuclear forces, and in unifying them with electromagnetism through the Standard Model of particle physics.

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