Wikipedia:Reference desk/Archives/Mathematics/2021 July 21

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July 21

continuous matrices

Graham Farmelo's biography of PAM Dirac describes Dirac using "continuous matrices", i.e. matrices indexed by real numbers rather than integers. Wikipedia doesn't have an article about those. Is that something that we'd now call a Hilbert space operator? Are those usually thought of as having a discrete (though infinite) set of basis vectors, and if yes, is the continuous version somehow the same? Should I study some functional analysis to understand this? I don't know any right now but feel like it could help. Thanks. 2601:648:8202:350:0:0:0:2B99 (talk) 09:12, 21 July 2021 (UTC)

I guess you mean Integral linear operator (the name "continuous matrices" is not used in Mathematics). pma 19:30, 21 July 2021 (UTC)

Thanks but it looks like integral linear operators are actually bilinear? I think of matrices as representations of linear operators. Unfortunately it is hard to figure out which specific Dirac paper Farmelo is referring to with that description, but I'll try reading more closely, or look at Dirac's book The Principles of QM. Do you happen to have a view of whether that book is still worth reading? I don't really know any QM and would like to learn some. 2601:648:8202:350:0:0:0:2B99 (talk) 07:53, 22 July 2021 (UTC)

I assume that biography is the book The Strangest Man. I could not find the term "continuous matrices" used in the book.  --Lambiam 12:20, 22 July 2021 (UTC)

In QM, the matrix representation of an operator T in some basis is given by the matrix elements $T_{ij}=\langle i|T|j\rangle$, where $|i\rangle$ and $|j\rangle$ are two basis vectors of the Hilbert space. When the Hilbert space is finite dimensional, i.e. when there's a finite number of basis states of the system (e.g. the two basis states of a spin 1/2 particle), the $T_{ij}$ form an ordinary matrix (this is basically Heisenberg's original formulation of QM). But there are situations where the state space has infinitely many possible states, and where these are continuous. The classical example is the free particles whose basis states can be given by either its location $|x\rangle$ or its momentum $|p\rangle$. In that case (whose development is largely due to Dirac if I'm not mistaken) one can still define "matrix elements" $\langle x|T|x'\rangle$ but now the "matrix" has infinitely (and uncountably) many components; it is actually a continuous function of two arguments, $T(x,x')$. I guess that this is what Farmelo means by "continous matrix". --Wrongfilter (talk) 13:18, 22 July 2021 (UTC)

Lambiam - whoops, sorry, I got another book confused with Farmelo's. I meant "Dirac: A Scientific Biography" by Helge Kragh. The part about continuous matrices is on page 40 and it appears to refer to Dirac's 1927 paper "The Physical Interpretation of the Quantum Mechanics", Proc. Royal Soc. of London, A113.

Wrongfilter, thanks, I'll try to understand that. I presume <x|T|x'> in math notation would be written <x, Tx'> where <·,·> is the inner product on the Hilbert space. I didn't realize QM used Hilbert spaces of uncountable dimension, or maybe I'm still confused on this issue. Do you know if Dirac's formulation was basically equivalent to von Neumann's from the same era, other than the terminology used? 2601:648:8202:350:0:0:0:2B99 (talk) 21:43, 22 July 2021 (UTC)