This lecture offers a conceptual and mathematical journey through the major pillars of 20th-century physics — general relativity, quantum mechanics, particle physics and cosmology — emphasizing the philosophical questions and unifying principles that connect them.

We begin with Einstein’s theory of general relativity, highlighting its geometric interpretation of gravity and its roots in the equivalence principle and Machian relationalism. From there, we transition into the quantum realm, contrasting the Schrödinger and Heisenberg formulations and arriving at Dirac’s abstract operator framework. The quantum harmonic oscillator serves as a unifying example across all three pictures.

Building on this foundation, we explore the emergence of second quantization, where fields — not particles — become the fundamental entities of nature. We trace how the algebra of creation and annihilation operators naturally leads to the concept of Fock space and the idea of particles as excitations of quantized fields.

Symmetries are introduced as the organizing principle behind all conservation laws and quantum numbers. From internal gauge symmetries (U(1), SU(2), SU(3)) to Lorentz invariance, we show how group representations give rise to the full zoo of elementary particles.

Special focus is given to the Dirac equation — not only as a relativistic wave equation for spin-1/2 particles, but also as the first theoretical framework to predict the existence of antimatter.

We conclude by connecting this formal structure to cosmology, examining how quantum fields and their symmetries shaped the early universe, influenced structure formation, and remain central to modern physical understanding.