Book: Lectures on Quantum Mechanics

Overview
Paul Dirac's Lectures on Quantum Mechanics (1966) presents a compact, rigorous course drawn from his lectures at Yeshiva University and elsewhere. The text addresses quantum theory at an advanced level, favoring algebraic reasoning and the Hamiltonian formalism over heuristic or pictorial approaches. Emphasis falls on the mathematical structure that underlies quantization and on how symmetry and constraints shape the transition from classical to quantum descriptions.
Dirac writes with characteristic economy and precision, guiding readers from classical mechanics to quantum operators while keeping the logical architecture transparent. The treatment expects mathematical maturity and a familiarity with canonical classical mechanics, but rewards the reader with a clear exposition of ideas that remained central to later developments in gauge theory and constrained quantization.

Formalism and methods
A key theme is the passage from Poisson brackets in classical mechanics to commutators in quantum mechanics, presented as an algebraic replacement rule grounded in the canonical formalism. Dirac systematically develops the Hamiltonian approach, canonical transformations, and the action principle, then formulates quantization rules in terms of operator algebra and transformation theory. The work stresses the primacy of operators and their algebraic relations over coordinate representations, making symmetry operations and conserved quantities particularly transparent.
Transformation theory receives attention as a unifying language for quantum processes, connecting different representations and clarifying the role of state vectors and observables. Dirac's style highlights underlying structures: the commutator algebra, the role of complex amplitudes, and the formal manipulations that lead to quantum dynamical equations.

Constrained systems and gauge theories
The most distinctive contribution is the systematic treatment of constrained Hamiltonian systems. Dirac introduces the classification of constraints into "first-class" and "second-class" and formulates the Dirac bracket to handle the latter in quantization. First-class constraints are shown to generate gauge transformations, and the interplay between constraints, gauge invariance, and physical degrees of freedom is analyzed with care.
Applications to electromagnetism and relativistic systems illustrate how constraints appear naturally and how they must be handled to obtain consistent quantum descriptions. The methods give a general prescription for quantizing systems with redundant variables, laying conceptual groundwork that later developments in gauge field quantization, such as Faddeev–Popov and BRST techniques, would build upon.

Examples and applications
Concrete examples punctuate the formal development: the simple harmonic oscillator and canonical quantization provide base cases, while relativistic particles and the electromagnetic field demonstrate how constraints and gauge conditions are implemented. Scattering theory and aspects of field quantization appear where needed to show how operator methods translate into physical predictions, including commutation relations for fields and the role of conserved currents.
Dirac's exposition keeps computations economical but complete, offering enough worked examples to illuminate subtleties without diluting the focus on structural principles. Readers encounter both finite-dimensional mechanical systems and infinite-dimensional field systems treated through the same consistent algebraic lens.

Style and legacy
The tone is terse and mathematically disciplined: proofs and derivations are concise, and physical interpretation is often stated with minimal adverbial commentary. The result is a text that rewards close reading and reflection rather than casual consultation. Its focus on algebraic methods and constrained dynamics made it highly influential for theorists working on gauge theories, quantization of fields, and the foundations of quantum mechanics.
Dirac's Lectures on Quantum Mechanics remains valued for its clarity of thought and conceptual precision. It stands as a classic reference for anyone seeking a deep understanding of how canonical formalism and symmetry considerations shape quantum theory, and as an historical bridge between early operator methods and modern approaches to gauge field quantization.