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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the scaling dimension, or simply dimension, of a local operator in a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
is
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
, scaling dimensions of operators are fixed numbers, otherwise they are functions of the distance scale.


Scale-invariant quantum field theory

In a
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, by definition each operator ''O'' acquires under a dilation x\to \lambda x a factor \lambda^, where \Delta is a number called the scaling dimension of ''O''. This implies in particular that the two point
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables rep ...
\langle O(x) O(0)\rangle depends on the distance as (x^2)^. More generally, correlation functions of several local operators must depend on the distances in such a way that \langle O_1(\lambda x_1) O_2(\lambda x_2)\ldots\rangle= \lambda^\langle O_1(x_1) O_2(x_2)\ldots\rangle Most scale invariant theories are also
conformally invariant In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
, which imposes further constraints on correlation functions of local operators.


Free field theories

Free theories are the simplest scale-invariant quantum field theories. In free theories, one makes a distinction between the elementary operators, which are the fields appearing in the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, and the composite operators which are products of the elementary ones. The scaling dimension of an elementary operator ''O'' is determined by dimensional analysis from the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
(in four spacetime dimensions, it is 1 for elementary bosonic fields including the vector potentials, 3/2 for elementary fermionic fields etc.). This scaling dimension is called the classical dimension (the terms canonical dimension and engineering dimension are also used). A composite operator obtained by taking a product of two operators of dimensions \Delta_1 and \Delta_2 is a new operator whose dimension is the sum \Delta_1+\Delta_2. When interactions are turned on, the scaling dimension receives a correction called the anomalous dimension (see below).


Interacting field theories

There are many scale invariant quantum field theories which are not free theories; these are called interacting. Scaling dimensions of operators in such theories may not be read off from a
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
; they are also not necessarily (half)integer. For example, in the scale (and conformally) invariant theory describing the critical points of the two-dimensional
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
there is an operator \sigma whose dimension is 1/8.In the
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
nomenclature, this theory is the minimal model M_ which contains the operators \sigma=\phi_ and \epsilon=\phi_.
Operator multiplication is subtle in interacting theories compared to free theories. The
operator product expansion In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vertex ...
of two operators with dimensions \Delta_1 and \Delta_2 will generally give not a unique operator but infinitely many operators, and their dimension will not generally be equal to \Delta_1+\Delta_2. In the above two-dimensional Ising model example, the operator product \sigma \times\sigma gives an operator \epsilon whose dimension is 1 and not twice the dimension of \sigma.


Non scale-invariant quantum field theory

There are many quantum field theories which, while not being exactly scale invariant, remain approximately scale invariant over a long range of distances. Such quantum field theories can be obtained by adding to free field theories interaction terms with small dimensionless couplings. For example, in four spacetime dimensions one can add quartic scalar couplings, Yukawa couplings, or gauge couplings. Scaling dimensions of operators in such theories can be expressed schematically as \Delta=\Delta_0 + \gamma(g), where \Delta_0 is the dimension when all couplings are set to zero (i.e. the classical dimension), while \gamma(g) is called the anomalous dimension, and is expressed as a power series in the couplings collectively denoted as g. Such a separation of scaling dimensions into the classical and anomalous part is only meaningful when couplings are small, so that \gamma(g) is a small correction. Generally, due to quantum mechanical effects, the couplings g do not remain constant, but vary (in the jargon of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, ''run'') with the distance scale according to their
beta-function In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is ...
. Therefore the anomalous dimension \gamma(g) also depends on the distance scale in such theories. In particular correlation functions of local operators are no longer simple powers but have a more complicated dependence on the distances, generally with logarithmic corrections. It may happen that the evolution of the couplings will lead to a value g=g_* where the
beta-function In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory. It is ...
vanishes. Then at long distances the theory becomes
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
, and the anomalous dimensions stop running. Such a behavior is called an
infrared fixed point In physics, an infrared fixed point is a set of coupling constants, or other parameters, that evolve from initial values at very high energies (short distance) to fixed stable values, usually predictable, at low energies (large distance). This usu ...
. In very special cases, it may happen when the couplings and the anomalous dimensions do not run at all, so that the theory is scale invariant at all distances and for any value of the coupling. For example, this occurs in the N=4 supersymmetric Yang–Mills theory.


References

Conformal field theory Quantum field theory {{Quantum-stub