Here is the section on Hilbert's work on Physics from the Wikipedia article "David Hilbert" At the end of the article, the Book "Mathematical Methods of Physics" by Courant and Hilbert is mentioned . It is an attempt to bring Mathematics and Physics together as is explicitly stated in the introduction.
"Physics[edit]
Until 1912, Hilbert was almost exclusively a "pure" mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend
Hermann Minkowski joked he had to spend 10 days in quarantine before being able to visit Hilbert. In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar in the subject in 1905.
In 1912, three years after his friend's death, Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself.
[37] He started studying
kinetic gas theory and moved on to elementary
radiation theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of
Albert Einstein and others were followed closely.
By 1907 Einstein had framed the fundamentals of the theory of gravity, but then struggled for nearly 8 years with a confounding problem of putting the theory into final form.
[38] By early summer 1915, Hilbert's interest in physics had focused on
general relativity, and he invited Einstein to Göttingen to deliver a week of lectures on the subject.
[39] Einstein received an enthusiastic reception at Göttingen.
[40] Over the summer Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts. During November 1915 Einstein published several papers culminating in "The Field Equations of Gravitation" (see
Einstein field equations).
[41] Nearly simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see
Einstein–Hilbert action). Hilbert fully credited Einstein as the originator of the theory, and no public priority dispute concerning the field equations ever arose between the two men during their lives.
[42] See more at
priority.
Additionally, Hilbert's work anticipated and assisted several advances in the
mathematical formulation of quantum mechanics. His work was a key aspect of
Hermann Weyl and
John von Neumann's work on the mathematical equivalence of
Werner Heisenberg's
matrix mechanics and
Erwin Schrödinger's
wave equation and his namesake
Hilbert space plays an important part in quantum theory. In 1926 von Neumann showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.
[43]
Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher math, physicists tended to be "sloppy" with it. To a "pure" mathematician like Hilbert, this was both "ugly" and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of
integral equations. When his colleague
Richard Courant wrote the now classic
Methoden der mathematischen Physik (Methods of Mathematical Physics) including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant-Hilbert book made it easier for them"