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In Sakurai's Quantum Mechanics the concept of a Hilbert space underlying classical quantum wave mechanics (Schrödinger equation) is extensively developed.

But when dealing with the Dirac equation this concept is dropped and one only works in the position basis (wave mechanics). Why is it that we never use the abstract bra-ket-formulation when dealing with relativistic QM?

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    $\begingroup$ Related: Dirac Equation in RQM (as opposed to QFT) is written in which representation? $\endgroup$
    – Alfred Centauri
    Commented Jun 9, 2020 at 16:33
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    $\begingroup$ To be clear: Are you inferring from the lack of such notation in Sakurai's section on the topic that "we" "never" use abstract bra-ket notation in rQM here? Or have you looked at other sources and also not found it there? One book is never representative of an entire field. $\endgroup$
    – ACuriousMind ♦
    Commented Jun 9, 2020 at 21:01
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    $\begingroup$ @ACuriousMind I read two lecture notes and this book. Of course this is not representative. Becaus of this I'm also very welcome with an answer like "You're wrong, heres such treatment, link" $\endgroup$
    – user224659
    Commented Jun 9, 2020 at 22:38

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I haven't read Sakurai so I cannot comment on it. However, bras, kets, Hilbert spaces and Hamiltonians are all perfectly relevant in quantum field theory which is relativistic. States are wave functionals. It's usually not written like this because it's clunky and not suited for most practical applications. However, it's often useful to think in terms of wave functionals to understand certain things like coherent states.

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One serious challenge in transitioning from "normal" quantum mechanics to relativistic quantum mechanics is that time and space have to be on the same footing. In nonrelativistic quantum mechanics, position is treated as an operator, while time is just a label on the wavefunction. One way to get around this problem is to demote position so that it is no longer an operator and merely plays the role of a label on the wavefunction.

Of course, that raises the question of which Hilbert space does the wavefunction occupy, since there is now no such thing as a position basis. The answer comes from quantum field theory. In quantum field theory, we abandon the concept of individual, isolated particles (after all, $E=mc^2$ implies new particles can be created out of energy). Instead, we treat fields as the fundamental objects and quantize them. What we end up with is an infinite number of degrees of freedom: one for each point in spacetime (the field can take any value at each point in spacetime). In "classical" quantum mechanics, there are three infinite degrees of freedom (for example, one for each spacial direction). In quantum field theory there are an infinite number of infinite degrees of freedom. The wavefunctions may be expanded in various bases belonging to infinite families of operators.

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One can derive the important results of relativistic quantum theory (or qft) without much using bra-ket notation. However, I think if one wants a deep mathematical understanding, it is better to use bra-ket notation throughout. I have done this in A Construction of Full QED Using Finite Dimensional Hilbert Space

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    $\begingroup$ Of what relevance is the link to your paper? When you cite yourself, please a) clearly state that you are doing so in the post itself and b) explain why it is relevant in the surrounding text. $\endgroup$
    – ACuriousMind ♦
    Commented Jun 9, 2020 at 20:57
  • $\begingroup$ @ACuriousMind, I have edited, but I thought it was already obvious. $\endgroup$
    – Charles Francis
    Commented Jun 9, 2020 at 21:00
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    $\begingroup$ What I thought was obvious was that you wish to promote your paper. I'd have found an explanation on why it is uniquely suited as a reference in this context (and not, say any other text that uses bra-ket notation in a relativistic context) re-assuring. $\endgroup$
    – ACuriousMind ♦
    Commented Jun 9, 2020 at 21:04
  • $\begingroup$ Okay I will take a look at this later. Right now this is on a level beyond what I'm looking for. $\endgroup$
    – user224659
    Commented Jun 9, 2020 at 22:39

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