nLab matrix mechanics

Idea

What is historically called the matrix mechanics formulation of quantum mechanics originates with Heisenberg 1925 and is therefore also known as the Heisenberg picture of quantum mechanics. Here algebras of linear operators (naively: “matrices”) represent quantum observables, and their non-trivial commutators reflect the uncertainty principle due to Heisenberg 1927.

The complementary perspective is that of “wave mechanics”, due to Schrödinger 1926, now also called the Schrödinger picture of quantum mechanics.

Over exotic ground rigs

To some extent, classical mechanics may be understood as a matrix mechanics akin to the situation in quantum mechanics, but with the ground rig $R$ taken not to be that of the complex number $R \equiv \mathbb{C}$, but the tropical semiring $R \equiv \mathbb{R}_{min} \coloneqq \mathbb{R} \cup \{+\infty\}$, where the addition is $min$ and the multiplication is $+$.

Similarly, statistical mechanics, or rather thermal statics, may be understood as matrix mechanics over a rig $\mathbb{R}^T$ that depends on a temperature parameter $T$. As $T \to 0$, the rig $\mathbb{R}^T$ reduces to $\mathbb{R}_{min}$ and thermal statics reduces to classical statics, just as quantum dynamics reduces to classical dynamics as Planck's constant approaches zero.

References

General

The origin of the “matrix mechanics” formulation of quantum mechanics is:

• Werner Heisenberg: Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik 33 (1925) 879–893 [doi:10.1007/BF01328377, Engl. pdf]

which however did not use the term “matrix” (which was not widely known then). The actual matrices in matrix mechanics were identified in:

• Max Born, Pascual Jordan: Zur Quantenmechanik, Zeitschrift für Physik 34 (1925) 858–888 [doi:10.1007/BF01328531]

• Max Born, Werner Heisenberg, Pascual Jordan: Zur Quantenmechanik. II., Z. Physik 35 (1926) 557–615 [doi:10.1007/BF01379806]

Original derivation of the uncertainty principle from non-trivial commutators of these “matrices”:

• Werner Heisenberg: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43 (1927) 172–198 [doi:10.1007/BF01397280, pdf]

  (where it is announced as equation (1) and established as equation (6))

with English translation (or something close) in

• The actual content of quantum theoretical kinematics and mechanics, NASA Technical Memorandum NAS 1.15:77379 (1983) [pdf, pdf]

Historical review:

• Marco Giliberti, Luisa Lovisetti: Matrix Mechanics, chapter in: Old Quantum Theory and Early Quantum Mechanics, Springer (2024) 397–429 [doi:10.1007/978-3-031-57934-9_11]

See also:

• Wikipedia: Matrix mechanics

Over other ground rigs

Discussion of classical/statistical mechanics as matrix mechanics over tropical ground rigs:

• Quantization and Cohomology: Week 12

• Quantization and Cohomology: Week 13

• superposition in classical mechanics

• Baez-Corfield discussion

• discussion continued - homotopy theory as groupoid valued mechanics

• Geometric Representation Theory (Lecture 12)