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We propose here a new symplectic quantization scheme, where quantumfluctuations of a scalar field theory stem from two main assumptions:relativistic invariance and equiprobability of the field configurations withidentical value of the action. In this approach the fictitious time ofstochastic quantization becomes a genuine additional time variable, withrespect to the coordinate time of relativity. This proper time is associated toa symplectic evolution in the action space, which allows one to investigate notonly asymptotic, i.e. equilibrium, properties of the theory, but also itsnon-equilibrium transient evolution. In this paper, which is the first one in aseries of two, we introduce a formalism which will be applied to generalrelativity in the companion work "Symplectic quantization II".

Symplectic quantization I: dynamics of quantum fluctuations in a relativistic field theory