13185 - Natural Sciences Methodology (1)

Learning outcomes

Two basic ingredients of quantum physics, i.e., complex numbers and probabilities, can be traced back to Renaissance. Is it just a matter of curiosity? No documented relation connects Renaissance painting and quantum view of the world, nevertheless they both need an "imaginary" dimension, entangled with the real one, to take shape.

This course is designed to provide an introduction to quantum physics from a "perspectival" point of view. The analysis of the symmetry and invariance principles, which rule the architecture of the physical world, will guide us to understand how complex numbers, involved in the description of quantum systems, change the notions of probability and indeterminism.

Course contents

SYMMETRIES, COMPLEX NUMBERS AND QUANTUM PHYSICS

• Geometry and arithmetic: the theorem of Pythagoras
• Regular and 'dependent' polyhendra
  
• Polygons and polynomial equations
• Complex numbers
• Bidimensional vector spaces
  
• States and observables in quantum mechanics
• Complex probability amplitudes
• Scalar product and projector operators
  
• Orthogonal and unitary transformations
  
• Symmetries and invariances

SEMINAR

Perspective symmetries and complex numbers between Renaissance and quantum physics

http://www.isa009.malakut.eu/lectures.htm

NB: Students are advised to attend classes regularly

Readings/Bibliography

R. Hughes, Structure and interpretation of quantum mechanics, Harvard Univ. Press, Cambridge Mass. 1989
T. Needham, Visual Complex Analysis, Clarendon Press, Oxford 1997 (cap. I)
V. Scarani, Quantum Physics. A First Encounter (2003), Oxford Univ. Press, Oxford 2006
J. Stillwell, Mathematics and Its History, Springer, New York 1989
H. Weyl, Symmetry, Princeton Univ. Press, Princeton 1952

Further reading

• I. Stewart, Why Beauty is Truth. A History of Symmetry, Basic Books, New York 2007
• B. van Fraassen, Laws and Symmetry, Clarendon Press, Oxford 1989
• B. van Fraassen, Quantum Mechanics, Clarendon Press, Oxford 1991
• H. Weyl, Philosophy of Mathematics and Natural Science, Princeton Univ. Press, Princeton 1949
  

Office hours

See the website of Rossella Lupacchini