# Does QFT modifies Quantum Mechanics? [duplicate]

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The basis of Quantum Mechanics is contained in the postulates which tell us how to describe quantum systems (below I disconsider possibly degenerate spectra just for simplicity):

1. To describe a quantum system there is a Hilbert Space $\mathcal{H}$ whose elements represent the possible states of the system.

2. For each physical quantity associated to a system there is one hermitian operator defined on $\mathcal{H}$. We call these operators observables.

3. The only possible values to be measured of an observable are the elements of its spectrum.

4. If $A$ is an observable with discrete spectrum $\sigma(A)=\{a_i : i \in \Bbb N\}$, then the probability of measuring $a_i$ on the state $|\psi\rangle$ is $P(a_i)=|\langle \varphi_i|\psi\rangle|^2$, where $|\varphi_i\rangle$ is the eigenstate corresponding to the eigenvalue $a_i$. Analogously, if $A$ is an observable with continuous spectrum $\sigma(A)$ then the probability density for the values of $A$ on the state $|\psi\rangle$ is $\rho(x)=|\langle x|\psi \rangle|^2$ where $x\in \sigma(A)$ and $|x\rangle$ is the generalized eigenvector corresponding to $x$.

5. When one performs a measurement of the observable $A$ the state colapses to the eigenstate corresponding to the eigenvalue measured.

6. The time evolution is governed by the requirement that the observable corresponding to the energy is the generator of time translations. That is, the time evolution equation is $i\hbar \frac{d|\psi\rangle}{dt}=H|\psi\rangle$.

Everything else follows from this. If we want to describe also spin we include Pauli's postulates and again, everything works just fine.

Now, apart from Quantum Mechanics there is Quantum Field Theory. I'm just starting to study it and there is something I still didn't get: Quantum Field Theory is just one application of Quantum Mechanics or it modifies Quantum Mechanics?

In other words, is QFT just one application of these postulates I've stated, or it modifies these postulates somehow? And if it does, how does QFT modifies Quantum Mechanics and its postulates?

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• QFT is QM. They just deal with different types of event. Someone more knowledgeable in the subject will answer you better. – QuantumBrick Jun 24 '16 at 4:10

## 2 Answers

Quantum field theory is based on quantum mechanics. The ground state which describes the fields is the free particle solution of the corresponding Dirac/KleinGordo/Maxwell equation.

QFT is a theory developed to be able to calculate the many body interactions, seen even in the simplest feynman diagram. It posits fields of each type of particle described by the above mentioned ground state , on which fields creation operators create a particle and annihilation destroy it, thus a particle is seen as a moving excitation on the field.

The basis is still on the quantum mechanics postulates. And its usefuleness lies in the one to one correlations with the feynman diagrams describing the interactions,

p.s. Field theories can be designed for other quantum mechanical many body problems; when I was a graduate student back in 1961 I was taught a field theory for nuclear physics states . It is a calculation tool in the end.

As far as I know, QFT does modify QM. Single particle description is problematic when you want to incooperate with Special Relativity, e.g. negative probability. Hence a quantum field theory is needed.

I recommend the book Quantum Field Theory for the Gifted Amateur, which is very useful for learning QFT.

• QFT is an application of quantum mechanics, but it doesn't modify it. All the relevant concepts stay in place. That non-relativistic quantum mechanics models have a different phenomenology is not a structural issue, they simply ignores the actually observed symmetry of the universe, whereas QFT doesn't. One can argue, though, that both QFT and non-relativistic are probably both toy-models of an even more self-consistent version of quantum mechanics which gets rid of a background spacetime. – CuriousOne Jun 24 '16 at 7:06