Provide a comprehensive analysis of the theoretical foundations of quantum mechanics. Discuss the historical development of quantum theory, its key principles (such as superposition, entanglement, and wave-particle duality), mathematical formulations (including Schrödinger's equation and Hilbert space), and major interpretations (Copenhagen, many-worlds, pilot-wave, etc.). Also examine the philosophical implications of quantum mechanics for our understanding of reality, measurement, and determinism. Finally, discuss how these theoretical foundations have led to technological applications and continue to influence modern physics.