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Contents
Context
Algebraic Quantum Field Theory
Physics
Contents
Idea
In physics, a scalar field is a field on spacetime/worldvolume which is simply a function with values in the field of scalars, typically the real numbers or complex numbers , sometimes the quaternions . Hence it is a field whose field bundle (of which the field is a section) is a trivial line bundle.
One fundamental (complex, charged) scalar field is seen in experiment, the Higgs field, which is one component of the standard model of particle physics. A widely hypothesized scalar field is the inflaton field in models of cosmic inflation, which however remains speculative and might in any case be an effective compound of more fundamental fields.
But scalar fields also serve as a key toy example in theoretical studies of field theory, such as in phi^4 theory or in the Ising model. The usefulness of the scalar field as a toy example of classical field theory and perturbative quantum field theory is due to it already exhibiting much of the core structure of field theory. For instance the general formulas for propagators and the S-matrix of general local field theories are structurally those of the scalar field, just with some more fairly evident representation theoretic structure thrown in.
Definition
Free scalar field
We discuss here the free scalar field on general spacetimes, hence the scalar field subject to the force of a background field of gravity, but not interacting with itself. This means that its local Lagrangian density is quadratic in the fields and its first derivatives (def. ) and its equation of motion is the Klein-Gordon equation (hence the wave equation in the case of vanishing mass) (prop. below).
The Poisson bracket on the covariant phase space of this system (prop. below) turns out to have as integral kernel the causal propagator of the Klein-Gordon operator (i.e. the Green function whose support is inside the light cone). Accordingly, the other associated Green functions of the Klein-Gordon operator (the “propagators”, such as the Feynman propagator) govern the perturbative quantum field theory of the scalar field (see at S-matrix for more).
Covariant phase space
Recall that a classical local field theory is for some prescribed class of manifolds of given dimension interpreted as spacetimes/worldvolumes:
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a choice of fiber bundle , called the field bundle;
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a choice of horizontal differential form of degree on the jet bundle of the field bundle, called the local Lagrangian density.
Given a classical local field theory defined by a local Lagrangian density , then
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the configuration space is the smooth space of sections of the field bundle;
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the equations of motion is the partial differential equation on elements given by
where
-
denotes the Euler-Lagrange operator
-
denotes jet prolongation.
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The covariant phase space is the subspace of solutions to the equations of motion, equipped with the canonical presymplectic form.
On Minkowski spacetime
We discuss the covariant phase space of the free scalar field on Minkowski spacetime. For a more detailed exposition see at geometry of physics – A first idea of quantum field theory.
Definition
(local Lagrangian density for free scalar field on Minkowski spacetime)
For , let spacetime be Minkowski spacetime of dimension , where denotes the Minkowski metric tensor of signature . We write for the corresponding volume form and for the canonical coordinate functions.
Let the field bundle be the trivial real line bundle over .
Then its jet bundle has canonical coordinates
In these coordinates, the local Lagrangian density
defining the free scalar field of mass on is
Proposition
(covariant phase space of free scalar field)
In the situation of def. for the -component of a section of the field bundle, its equation of motion is the Klein-Gordon equation
Moreover, the induced pre-symplectic current is, in local coordinates,
and hence the induced symplectic form on the covariant phase space of the free scalar field takes two smooth function , regarded as tangent vectors at zero to
where is any Cauchy surface and where denotes its time-like normal vector field.
Proof
We need to show that Euler-Lagrange operator takes the local Lagrangian density for the free scalar field to
First of all, the result of applying the vertical differential to the local Lagrangian density is
By definition of the Euler-Lagrange operator, in order to find and , we need to exhibit this as the sum of the form .
The key to find is to realize as a horizontal derivative. Since this is accomplished by
Hence we may set
because with this we have
In conclusion this yields the decomposition of the vertical differential of the Lagrangian density
which shows that is as claimed, and that is a presymplectic potential current. Hence the presymplectic current itself is
For a Cauchy surface, the transgression of this presymplectic current to the infinitesimal neighbourhood of is
Example
(Poisson brackets over Minkowski spacetime)
Consider the covariant phase space over Minkowski spacetime of dimension as in def. , with pre-symplectic current according to prop. given by
The corresponding Poisson bracket Lie (p+1)-algebra has in degree 0 Hamiltonian forms such as
and
The corresponding Hamiltonian vector fields are
and
Hence the corresponding bracket is
More generally for two bump functions then
Upon transgression to the Cauchy surface this yields the Poisson bracket
where now
are the point-evaluation functions (functionals), which act on a field configuration as
Notice that these point-evaluation functions themselves do not arise as the transgression of elements in , only their smearings such as do. Nevertheless we may express the above Poisson bracket conveniently via the integral kernel
(1)
More generally one may express the integral kernel for the Poisson bracket of evaluation functions for different values of . Notice that for each time interval we have a Lagrangian correspondence
where is the space of solutions to the Klein-Gordon equation on .
There is a unique function on whose pullback to is the evaluation function for any . By convenient abuse of notation, we also call that function .
Proposition
(integral kernel for Poisson bracket on Minkowski spacetime is the causal propagator)
The integral kernel on for the Poisson bracket of the scalar field over Minkowski spacetime (example ) is the causal propagator
(also known as the Pauli-Jordan distribution or Peierls bracket) on Minkowski spacetime:
(e. g. Scharf 01 (1.1.13))
Proof
By Fourier transform the general solution to the Klein-Gordon equation may be expressed in the form
where is the complex amplitude of the th mode .
We may split this into the contributions with positive and those with negative energy by decomposing the integral over as
By changing integration variables via this yields
where we defined the on-shell energy
It is convenient to also change variables in the second integral. This yields
Since is real-valued, it follows that under complex conjugation the amplitudes are related by
We abbreviate (cf. Scharf 01 (1.1.18))
where the prefactor just serves to make some of the following formulas come out conveniently.
With this the general solution to the Klein-Gordon equation is finally of the form
(2)
and hence its time derivative is
This allows to express the modes in terms of the value of the field and its time derivative at :
As in example we denote the corresponding evaluation functional
by the corresponding lower case symbol:
With the Poisson bracket kernel from example (1), it follow that the (integral kernel for the) Poisson bracket of these mode functionals is that of the canonical commutation relations:
(3)
where in the last step we used the Fourier transform representation of the delta distribution (this prop.).
In order to finally compute , it is convenient to break this up into two contributions: Write
for the positive and negative energy contributions from the Fourier expansion in (2), so that
Using the canonical commutation relation of the mode functions (3), we find
(4)
where in the last line we again applied change of integration variables. This is known as the 2-point function or Hadamard propagator on Minkowski spacetime (see def. below).
Similarly
In particular this says that
With this we finally obtain the expression for the causal propagator as the skew-symmetrization of the 2-point function:
We record the 2-point function that appeared in this computation:
On general spacetimes
We discuss the covariant phase space of the free scalar field on general spacetimes.
Definition
(local Lagrangian density for free scalar field on general spacetime)
As a classical local field the relativistic free scalar field in dimension of mass is
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the field bundle given by the trivial line bundle over ;
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the local Lagrangian density (a horizontal differential form on the jet bundle of the trivial line bundle over ) given by
where
-
denotes the norm square of the first order jets with respect to the given metric , hence in a local coordinate chart of the function
-
denotes the volume form of , canonically regarded as a horizontal differential form on .
The analogue of prop. holds true for general spacetimes:
(e.g. Bär-Ginoux-Pfäffle 07, corollary 3.4.3)
By this prop. (e.g. Khavkine 14, Collini 16, lemma 21). See also Fredenhagen-Rejzner 15, 3.3 Example
Interacting scalar field
We discuss perturbative quantum field theory of the free scalar field perturbed by an interaction term via locally covariant perturbative algebraic quantum field theory.
On Minkowski spacetime
We disscuss the interacting scalar field on Minkowski spacetime via causal perturbation theory.
By the discussion at S-matrix we need to determine the Feynman propagator, which is a linear combination of the 2-point function with the advanced propagator:
Definition
advanced propagator
retarded propagator
References
For instance
Most of the literatur on causal perturbation theory and perturbative AQFT focuses on the scalar field, for ease of exposition. See the references there.
The standard perturbative quantum field theory (made rigorous via causal perturbation theory) of the interacting scalar field is shown to be Fedosov deformation quantization of the corresponding covariant phase space in
For references on the construction of perturbative scalar field theory in causal perturbation theory see at locally covariant perturbative quantum field theory.
Discussion of scalar fields in cosmology includes
Examples in AQFT of non-perturbative interacting scalar field theory in any spacetime dimension (in particular in ) are claimed in