Index notation vs Dirac notation

[Thrice]

A professor of mine recently remarked that dirac notation is easily the best in physics & we'd quickly realize this once we take a course in relativity. I've already taken the course & I find myself disagreeing with him, but maybe that's only because I enjoy relativity more. Curious what you guys think.

 

[robphy]

What is he specifically referring to?
The notions of vectors and dual-vectors?

 

[Thrice]

He didn't specify & I just assumed the tensor notation. After seeing maxwell's equations reduced to 2 compact expressions, I'm having a hard time agreeing with him.

 

[coalquay404]

A professor of mine recently remarked that dirac notation is easily the best in physics & we'd quickly realize this once we take a course in relativity. I've already taken the course & I find myself disagreeing with him, but maybe that's only because I enjoy relativity more. Curious what you guys think.

There's nothing particularly great about Dirac's notation outside of, say, quantum mechanics and linear algebra. For example, given some vector space V(n,\mathbb{F}) with dual V^*(n,\mathbb{F}), we could choose to use Dirac's notation to denote an inner product between |\psi\rangle\in V(n,\mathbb{F}) and \langle\phi|\in V^*(n,\mathbb{F}) by

\langle\phi|\psi\rangle\in\mathbb{F}

However, in index notation you would choose a basis \{e_i\} for V(n,\mathbb{F}) and a corresponding dual basis \{\omega^i\} for V^*(n,\mathbb{F}). Then the inner product of the above quantities is

\phi_i\psi^i\in\mathbb{F}[/itex]<br /> <br /> Dirac&#039;s notation is fine for dealing with vectors and their duals. However, things quickly become cumbersome when you deal with tensor products. It&#039;s not uncommon for one to deal with tensors of rank four and above; in coordinate free notation this would simply be \mathbf{T}(W,X,Y,Z), while in index notation it&#039;s just T^{ijkl}. Contrast that with it&#039;s representation in Dirac notation:<br /> <br /> \mathbf{T}\to&lt;br /&gt; |W\rangle\otimes|X\rangle\otimes|Y\rangle\otimes|Z\rangle.<br /> <br /> See? It&#039;s too cumbersome to bother with. As with all of these kinds of things, notation is just a tool: you pick the one most suited to the job at hand.

 

[Thrice]

\phi_i\psi^i\in\mathbb{F}[/itex]

Now that&#039;s interesting. I remember doing tensors as operators, but I never thought of index notation in QM. I&#039;m guessing we mostly use the dirac notation because of historical reasons?

 

[coalquay404]

Now that's interesting. I remember doing tensors as operators, but I never thought of index notation in QM. I'm guessing we mostly use the dirac notation because of historical reasons?

I think it's a mixture of historical reasons (particularly Dirac's early writings, which are, by any standards, monumental) and geographic location. Anybody studying at UT, Austin probably uses (\phi,\psi)=(\psi,\phi)^* to denote inner products: it seems to be Weinberg's notation of choice.

Also, it's important to note the context in which Dirac's notation is particularly useful. In quantum mechanics, states are represented by rays in a complex separable Hilbert space. The operative word here is complex. In Dirac's notation, the inner product has the special property

\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*[/itex]<br /> <br /> where the star denotes complex conjugation. I&#039;m not sure if (a) it&#039;s really necessary to use Dirac notation in, say, general relativity since you rarely need to deal with complexified GR and (b) if it&#039;s really practical to deal with conjugation acting on multiple indices using stars.

 

[quantum123]

The Dirac bra ket notation is much more superior and neat in demonstating the dual space and vector space concept!

The super and sub-scripts are untidy, so much so that Penrose have to invent a string diagram where he draw strings to join all the indices.