The factor of 1/2 in certain mathematical expressions and formulas in quantum field theory is indeed a conventional choice that arises from various aspects of the theory, including group theory, Lie algebras, spinors, and physics itself. Let's explore some of the key reasons behind this convention:
Representations of Spinors: In quantum field theory, spinors describe the behavior of fermionic particles such as electrons. Spinors are mathematical objects that transform under certain symmetry groups, and their representations can be described by spinor fields. The factor of 1/2 appears in the transformation properties of spinor fields when they undergo a rotation by 2π. This arises due to the mathematics of spinor representations, and it leads to the appearance of 1/2 in various formulas involving spinors.
Symmetry Groups and Lie Algebras: Quantum field theories often involve symmetry groups, such as the Lorentz group or the gauge groups associated with fundamental forces. These symmetry groups are related to Lie algebras, which are mathematical structures that describe the generators of symmetries. In the process of quantizing a field theory, one typically expands the field operators in terms of creation and annihilation operators. The factors of 1/2 in certain commutation or anticommutation relations for these operators arise from the specific structure of the Lie algebra associated with the symmetry group.
Normalization of States: In quantum mechanics and quantum field theory, physical states are often normalized to have unit probability. The factor of 1/2 arises in the normalization of certain states, such as the normalization of two-particle states, which involves accounting for the indistinguishability of identical particles. The convention of 1/2 ensures that the probabilities and amplitudes calculated using these normalized states yield physically meaningful results.
Relativistic Quantum Mechanics: Relativistic quantum mechanics, which is incorporated into quantum field theory, introduces additional considerations related to the energy-momentum relation. The factor of 1/2 appears in certain formulas involving energy and momentum, such as the relation between energy and frequency (E = ħω) and the momentum of particles (p = ħk), where ħ is the reduced Planck constant. These formulas ensure consistency between quantum mechanics and relativity and are conventionally written with the factor of 1/2 for consistency and convenience.
It's important to note that the choice of the factor of 1/2 is not arbitrary but arises from the mathematical and physical structures inherent in quantum field theory. While it may seem purely conventional in some cases, it is deeply connected to the symmetries, representations, and principles that underlie the theory, ensuring its consistency and compatibility with experimental observations.