Classical scaling dimension

In theoretical physics, namely quantum field theory, the classical scaling dimension of an operator O is the power of mass of an operator determined by dimensional analysis from the Lagrangian (1 for elementary bosonic fields including the vector potentials, 3/2 for elementary fermionic fields etc.). It is also called the naïve dimension or the engineering dimension.

The dimension of a product is the sum of the dimensions. For example, in four-dimensional quantum field theory, the Lagrangian must have dimension of mass to the fourth power. By a dimension of the individual terms, we mean the dimension of the corresponding operators with the coefficients removed.

The operators whose coefficients are dimensionless are called marginal operators in the context of renormalization group or dimension four operators. The operators of dimension five are dimension five operators and much like other operators whose dimension exceeds four, they are irrelevant operators. Operators with dimensions lower than four, e.g. dimension three operators, are relevant operators.

See also

• Conformal weight
• Anomalous scaling dimension
• Dimensional analysis