1. The important role of functional integrals in modern physics J. Zinn-Justin CEA/IRFU (irfu.cea.fr), Centre de Saclay and Email: jean.zinn-justin@cea.fr

2. Abstract Twentieth century has seen the emergence of the physics of fluctuating systems, statistical and quantum. This explains, to some extent, the important role played by functional integrals in modern physics. We describe here a few striking examples of physics problems where the use of path or field integrals has proven decisive. 2 J Zinn-Justin, IPhT Saclay, 03/05/2011

3. For an introductory note about path integrals in physics see, for example, J. Zinn-Justin, Path integral, Scholarpedia, 4(2): 8674 (2009) (www.scholarpedia.org).www.scholarpedia.org For details and more references see, for example, and J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press 1989 (Oxford 4th ed. 2002). 3 J Zinn-Justin, IPhT Saclay, 03/05/2011

4. J Zinn-Justin, IPhT Saclay 03/05/2011 4 Classical equations: The mysterious variational principle

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12. Unitarity and covariance of the relativistic quantum field theory 12 J Zinn-Justin, IPhT Saclay, 03/05/2011

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14. PERTURBATION THEORY: FIELD INTEGRALS AS ALGEBRAIC TOOLS 14

15. Quantization of non-Abelian gauge theories 15 J Zinn-Justin, IPhT Saclay, 03/05/2011

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20. J Zinn-Justin,, IPhT Saclay, 03/05/2011 20 Quantization of the non-linear σ-model

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22. BEYOND PERTURBATION THEORY 22

23. Critical phenomena, quantum field theory and regularization 23 J Zinn-Justin, IPhT Saclay, 03/05/2011

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25. Numerical simulations in quantum field theory 25 J Zinn-Justin, IPhT Saclay, 03/05/2011

26. Instantons, barrier penetration and vacuum instability 26 J Zinn-Justin, IPhT Saclay, 03/05/2011

27. Instantons, large order behaviour and the problem of Borel summability 27 J Zinn-Justin, IPhT Saclay, 03/05/2011

28. Reliable critical exponents from O(N) symmetric (φ 2 ) 2 3 field theory (Le Guillou and Z.-J. (1980) updated by Guida and Z.-J. (1998)) 28 J Zinn-Justin, IPhT Saclay, 03/05/2011

29. Potentials with degenerate minima 29 J Zinn-Justin, IPhT Saclay, 03/05/2011

30. Relation between classical and quantum statistical physics 30 J Zinn-Justin, IPhT Saclay, 03/05/2011

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32. Large N non-perturbative techniques 32 J Zinn-Justin, IPhT Saclay, 03/05/2011