# What is the difference between quantum mechanics and QFT?

#### lonewolf219

Hi,

I'm an undergrad that has not taken quantum mechanics... What exactly is the distinction between quantum mechanics and quantum field theory? Does quantum mechanics describe the interactions of particles? Are Feynman diagrams used in quantum mechanics?

I have been doing some reading over the internet and everything is starting to blur together... Also, if anyone can explain why Lie Algebra is used, that would be much appreciated...and why we need the Pauli matrices to form an orthogonal basis (why orthogonal?) in Hilbert Space (why not Euclidean?)

Thanks...

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#### yenchin

For your main question, look at the first few pages of David Tong's QFT notes, under the section "Why Quantum Field Theory". Hope that helps.

#### lonewolf219

Thank you, I just took a quick look and read one sentence that is very helpful..

"The primary reason for introducing the concept of the field is to construct laws of Nature which are local"...

Appreciate it!

#### tiny-tim

Homework Helper
hi lonewolf219!
What exactly is the distinction between quantum mechanics and quantum field theory? Does quantum mechanics describe the interactions of particles? Are Feynman diagrams used in quantum mechanics?
QM is physics (including the maths tools necessary for the physics).

QFT is a maths tool.

Feynman diagrams are only used as part of QFT.
Also, if anyone can explain why Lie Algebra is used, that would be much appreciated...and why we need the Pauli matrices to form an orthogonal basis (why orthogonal?) in Hilbert Space (why not Euclidean?)
In QM, everything has an amplitude.

The probability is the amplitude times its complex conjugate.

That means the amplitudes must be in a space with an inner product and a complex conjugate … we call that a Hilbert space.

The different basis elements of a Hilbert space must be orthogonal (ie have inner product zero with each other). Making calculations would be impossibly long if we didn't express everything in terms of basis elements, that's why we need them.

#### DarMM

Gold Member
Hi,

I'm an undergrad that has not taken quantum mechanics... What exactly is the distinction between quantum mechanics and quantum field theory?
Quantum field theory refers to several different theories were objects called "fields", obeying
the laws of quantum mechanics, are used to model particle interactions.

Quantum Mechanics can refer to any quantum theory. Sometimes it is used to specifically
mean quantum theories which involve a fixed number of particles interacting with various classical potential energy distributions.

For example, you can make a quantum mechanical model of hydrogen. Here the electron is quantum mechanical and the coloumb potential is classical.

In a quantum field theory model of hydrogen, the electron is just a part of the electron field and the coloumb potential also results from the quantised photon field.

Does quantum mechanics describe the interactions of particles? Are Feynman diagrams used in quantum mechanics?
Yes, to the first question. For the second question, yes they are, but not as commonly as the would be in quantum field theory.

I have been doing some reading over the internet and everything is starting to blur together... Also, if anyone can explain why Lie Algebra is used, that would be much appreciated...
Lie algebras are just a certain type of algebra (vector spaces with a multiplication rule). They are important becuase they pop up in studying symmetries in quantum theories.

and why we need the Pauli matrices to form an orthogonal basis (why orthogonal?)
The Pauli matricies are related to angular momentum, we need them when studying the spin of particles. We would want an orthogonal basis because it's mathematically much simpler, it seperates angular momentum into three mutually perpendicular directions.

in Hilbert Space (why not Euclidean?)
Quantum Mechanics uses Hilbert spaces, not Euclidean spaces.

#### lonewolf219

Great!!! Thank you very much! I wish I asked earlier!

#### dextercioby

Homework Helper
I was too about to say something, but DarMM said it all.

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