Where exactly does QFT differ from QM? (in their formalisms)

  • I
  • Thread starter carllacan
  • Start date
  • Tags
    Qft Qm
In summary, QFT is a generalization of quantum mechanics that incorporates special relativity and allows for the creation and destruction of particles. It is based on the concept of fields rather than individual particles and involves the quantization of these fields. QFT can be used in both relativistic and non-relativistic physics and is represented by a direct sum of Hilbert spaces for different numbers of particles. The key difference between QFT and quantum mechanics is the algebra of observables, with QFT involving fields and quantum mechanics involving point particles.
  • #36
The mathematical model is quite different so it is not easy to explain how QM and QFT differ in a post. I'll try giving the general change of picture in a few strokes.
The main difference is that one goes from a Hilbert space of functions acted by operators(observables) that is suited for imagining the wavefunction somewhat as belonging to a quantum particle, to the concept of quantum field as a distribution whose values are operators (functionals) acting on states in a Fock space. These operators can be thought of (if one continues with the analogy of wavefunctions of particles in QM) as creating particles(in the case of creation operators) from the vacuum state they act on. So one can see the mathematical model changes quite a bit, from an infinite-dimensional Hilbert space with finite degrees of freedom to an infinite dimensional Fock space with infinite degrees of freedom and that allows changing numbers of particles.
All this works fine mathematically as long as you only consider free fields. With interacting fields there is no longer well defined Fock space (as mentioned in another post) due to certain theorem and QFT is just an extremely useful heuristic and predictive calculational tool.
 
Last edited:
<h2> What is the main difference between the formalisms of QFT and QM?</h2><p>The main difference between the formalisms of QFT (Quantum Field Theory) and QM (Quantum Mechanics) is that QFT is a quantum theory that describes the behavior of particles as fields in space and time, while QM describes the behavior of individual particles in terms of wavefunctions.</p><h2> How do the mathematical equations in QFT differ from those in QM?</h2><p>The mathematical equations in QFT are more complex and involve fields and operators, while the equations in QM are simpler and involve wavefunctions and operators. QFT also incorporates special relativity into its equations, while QM does not.</p><h2> What is the role of symmetry in QFT and QM?</h2><p>Symmetry plays a crucial role in both QFT and QM. In QFT, symmetry principles such as gauge symmetry and Lorentz symmetry are used to construct the theory and make predictions. In QM, symmetry principles such as rotational symmetry and time translation symmetry are used to determine the properties of particles.</p><h2> How does the concept of particles differ in QFT and QM?</h2><p>In QFT, particles are described as excitations of fields, while in QM, particles are described as wave-like entities. In QFT, particles can also be created and destroyed, whereas in QM, particles are considered to be fundamental and cannot be created or destroyed.</p><h2> What are the practical applications of QFT and QM?</h2><p>QFT has been successfully applied to describe the behavior of subatomic particles, such as in the Standard Model of particle physics. QM has been applied to various fields, including chemistry, materials science, and electronics. Both theories have also been used in the development of technologies such as transistors and lasers.</p>

FAQ: Where exactly does QFT differ from QM? (in their formalisms)

What is the main difference between the formalisms of QFT and QM?

The main difference between the formalisms of QFT (Quantum Field Theory) and QM (Quantum Mechanics) is that QFT is a quantum theory that describes the behavior of particles as fields in space and time, while QM describes the behavior of individual particles in terms of wavefunctions.

How do the mathematical equations in QFT differ from those in QM?

The mathematical equations in QFT are more complex and involve fields and operators, while the equations in QM are simpler and involve wavefunctions and operators. QFT also incorporates special relativity into its equations, while QM does not.

What is the role of symmetry in QFT and QM?

Symmetry plays a crucial role in both QFT and QM. In QFT, symmetry principles such as gauge symmetry and Lorentz symmetry are used to construct the theory and make predictions. In QM, symmetry principles such as rotational symmetry and time translation symmetry are used to determine the properties of particles.

How does the concept of particles differ in QFT and QM?

In QFT, particles are described as excitations of fields, while in QM, particles are described as wave-like entities. In QFT, particles can also be created and destroyed, whereas in QM, particles are considered to be fundamental and cannot be created or destroyed.

What are the practical applications of QFT and QM?

QFT has been successfully applied to describe the behavior of subatomic particles, such as in the Standard Model of particle physics. QM has been applied to various fields, including chemistry, materials science, and electronics. Both theories have also been used in the development of technologies such as transistors and lasers.

Similar threads

A Importance of Densities in QFT
Replies
1
Views
1K
A What is the relationship between QFT and QM?
Replies
34
Views
4K
I Where does the QM creation operator get its energy?
Replies
29
Views
2K
A QM is Feynman path ensemble - is QFT Feynman field ensemble?
3
Replies
89
Views
4K
A How do entanglement experiments benefit from QFT (over QM)?
14
Replies
473
Views
26K
B Questions on QFT & QM: Is QM or QFT Absolute Time?
2
Replies
69
Views
5K
I Hermitian operators in QM and QFT
Replies
7
Views
2K
I How to get the wavefunction of a single particle in QFT?
Replies
2
Views
2K
B QFT & QM Questions (about time, locality, gravity, etc.)
Replies
6
Views
2K
A Measurement in QFT: Mapping Fields to Theory's Math Formalism
Replies
31
Views
2K

Hot Threads

Recent Insights

Back
Top