In
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 ...
, scalar field theory can refer to a relativistically invariant
classical or
quantum theory of
scalar fields. A scalar field is invariant under any
Lorentz transformation.
The only fundamental scalar quantum field that has been observed in nature is the
Higgs field
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the St ...
. However, scalar quantum fields feature in the
effective field theory
In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees ...
descriptions of many physical phenomena. An example is the
pion, which is actually a
pseudoscalar.
Since they do not involve
polarization complications, scalar fields are often the easiest to appreciate
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
through. For this reason, scalar field theories are often used for purposes of introduction of novel concepts and techniques.
The
signature of the metric employed below is .
Classical scalar field theory
A general reference for this section is Ramond, Pierre (2001-12-21). Field Theory: A Modern Primer (Second Edition). USA: Westview Press. , Ch 1.
Linear (free) theory
The most basic scalar field theory is the
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
theory. Through the
Fourier decomposition
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
of the fields, it represents the
normal modes
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies ...
of an
infinity of coupled oscillators where the
continuum limit of the oscillator index ''i'' is now denoted by . The
action for the free
relativistic scalar field theory is then
:
where
is known as a
Lagrangian density; for the three spatial coordinates; is the
Kronecker delta function; and for the -th coordinate .
This is an example of a quadratic action, since each of the terms is quadratic in the field, . The term proportional to is sometimes known as a mass term, due to its subsequent interpretation, in the quantized version of this theory, in terms of particle mass.
The equation of motion for this theory is obtained by
extremizing the action above. It takes the following form, linear in ,
:
where ∇
2 is the
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
. This is the
Klein–Gordon equation, with the interpretation as a classical field equation, rather than as a quantum-mechanical wave equation.
Nonlinear (interacting) theory
The most common generalization of the linear theory above is to add a
scalar potential to the Lagrangian, where typically, in addition to a mass term, ''V'' is a polynomial in . Such a theory is sometimes said to be
interacting, because the Euler–Lagrange equation is now nonlinear, implying a
self-interaction
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
. The action for the most general such theory is
:
The ''n''! factors in the expansion are introduced because they are useful in the Feynman diagram expansion of the quantum theory, as described below.
The corresponding Euler–Lagrange equation of motion is now
:
Dimensional analysis and scaling
Physical quantities in these scalar field theories may have dimensions of length, time or mass, or some combination of the three.
However, in a relativistic theory, any quantity , with dimensions of time, can be readily converted into a ''length'', , by using the
velocity of light, . Similarly, any length is equivalent to an inverse mass, , using
Planck's constant, . In natural units, one thinks of a time as a length, or either time or length as an inverse mass.
In short, one can think of the dimensions of any physical quantity as defined in terms of ''just one'' independent dimension, rather than in terms of all three. This is most often termed the
mass dimension of the quantity. Knowing the dimensions of each quantity, allows one to ''uniquely restore'' conventional dimensions from a natural units expression in terms of this mass dimension, by simply reinserting the requisite powers of and required for dimensional consistency.
One conceivable objection is that this theory is classical, and therefore it is not obvious how Planck's constant should be a part of the theory at all. If desired, one could indeed recast the theory without mass dimensions at all: However, this would be at the expense of slightly obscuring the connection with the quantum scalar field. Given that one has dimensions of mass, Planck's constant is thought of here as an essentially ''arbitrary fixed reference quantity of action'' (not necessarily connected to quantization), hence with dimensions appropriate to convert between mass and
inverse length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre (symbol: m&minus ...
.
Scaling dimension
The
classical scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is scale ...
, or mass dimension, , of describes the transformation of the field under a rescaling of coordinates:
:
:
The units of action are the same as the units of , and so the action itself has zero mass dimension. This fixes the scaling dimension of the field to be
:
Scale invariance
There is a specific sense in which some scalar field theories are
scale-invariant. While the actions above are all constructed to have zero mass dimension, not all actions are invariant under the scaling transformation
:
:
The reason that not all actions are invariant is that one usually thinks of the parameters ''m'' and as fixed quantities, which are not rescaled under the transformation above. The condition for a scalar field theory to be scale invariant is then quite obvious: all of the parameters appearing in the action should be dimensionless quantities. In other words, a scale invariant theory is one without any fixed length scale (or equivalently, mass scale) in the theory.
For a scalar field theory with spacetime dimensions, the only dimensionless parameter satisfies = . For example, in = 4, only is classically dimensionless, and so the only classically scale-invariant scalar field theory in = 4 is the massless
4 theory.
Classical scale invariance, however, normally does not imply quantum scale invariance, because of the
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in t ...
involved – see the discussion of the beta function below.
Conformal invariance
A transformation
:
is said to be
conformal if the transformation satisfies
:
for some function .
The conformal group contains as subgroups the
isometries of the metric
(the
Poincaré group) and also the scaling transformations (or
dilatation
Dilation (or dilatation) may refer to:
Physiology or medicine
* Cervical dilation, the widening of the cervix in childbirth, miscarriage etc.
* Coronary dilation, or coronary reflex
* Dilation and curettage, the opening of the cervix and surgi ...
s) considered above. In fact, the scale-invariant theories in the previous section are also conformally-invariant.
4 theory
Massive
4 theory illustrates a number of interesting phenomena in scalar field theory.
The Lagrangian density is
:
Spontaneous symmetry breaking
This Lagrangian has a ℤ₂ symmetry under the transformation .
This is an example of an
internal symmetry
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
A family of particular transformations may be ''continuo ...
, in contrast to a
space-time symmetry.
If is positive, the potential
:
has a single minimum, at the origin. The solution ''φ''=0 is clearly invariant under the ℤ₂ symmetry.
Conversely, if is negative, then one can readily see that the potential
:
has two minima. This is known as a ''double well potential'', and the lowest energy states (known as the vacua, in quantum field theoretical language) in such a theory are invariant under the ℤ₂ symmetry of the action (in fact it maps each of the two vacua into the other). In this case, the ℤ₂ symmetry is said to be ''
spontaneously broken
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
''.
Kink solutions
The
4 theory with a negative
2 also has a kink solution, which is a canonical example of a
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
. Such a solution is of the form
: