Covariant Hamiltonian Field Theory
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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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, Hamiltonian field theory is the field-theoretic analogue to classical
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
. It is a formalism in
classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...
alongside Lagrangian field theory. It also has applications in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.


Definition

The
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
for a system of discrete particles is a function of their
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...
and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field. Since each point mass has one or more
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
, the field formulation has infinitely many degrees of freedom.


One scalar field

The Hamiltonian density is the continuous analogue for fields; it is a function of the fields, the conjugate "momentum" fields, and possibly the space and time coordinates themselves. For one
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
, the Hamiltonian density is defined from the
Lagrangian density Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees ...
byIt is a standard abuse of notation to abbreviate all the derivatives and coordinates in the Lagrangian density as follows: :\mathcal (\phi, \partial_\mu \phi, x_\mu) The is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In general, all the spatial and time derivatives will appear in the Lagrangian density, for example in Cartesian coordinates, the Lagrangian density has the full form: :\mathcal \left(\phi, \frac, \frac, \frac, \frac, x,y,z,t\right) Here we write the same thing, but using ∇ to abbreviate all spatial derivatives as a vector. :\mathcal(\phi, \pi, \mathbf,t) = \dot\pi - \mathcal(\phi, \nabla\phi, \partial \phi/\partial t , \mathbf,t)\,. with the "del" or "nabla" operator, is the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
of some point in space, and is
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
. The Lagrangian density is a function of the fields in the system, their space and time derivatives, and possibly the space and time coordinates themselves. It is the field analogue to the Lagrangian function for a system of discrete particles described by generalized coordinates. As in Hamiltonian mechanics where every generalized coordinate has a corresponding generalized momentum, the field has a conjugate momentum field , defined as the partial derivative of the Lagrangian density with respect to the time derivative of the field, :\pi = \frac \,,\quad \dot\equiv\frac\,, in which the overdotThis is standard notation in this context, most of the literature does not explicitly mention it is a partial derivative. In general total and partial time derivatives of a function are not the same. denotes a ''partial'' time derivative , not a
total Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are comp ...
time derivative .


Many scalar fields

For many fields and their conjugates the Hamiltonian density is a function of them all: :\mathcal(\phi_1, \phi_2, \ldots, \pi_1, \pi_2, \ldots, \mathbf,t)= \sum_i\dot\pi_i - \mathcal(\phi_1,\phi_2,\ldots \nabla\phi_1,\nabla\phi_2,\ldots, \partial \phi_1/\partial t ,\partial \phi_2/\partial t ,\ldots, \mathbf,t)\,. where each conjugate field is defined with respect to its field, :\pi_i(\mathbf,t) = \frac \,. In general, for any number of fields, the
volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applica ...
of the Hamiltonian density gives the Hamiltonian, in three spatial dimensions: :H = \int \mathcal \ d^3 x \,. The Hamiltonian density is the Hamiltonian per unit spatial volume. The corresponding
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
is nergylength]−3, in
SI units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
Joules per metre cubed, J m−3.


Tensor and spinor fields

The above equations and definitions can be extended to
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s and more generally
tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
s and
spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\co ...
s. In physics, tensor fields describe
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s and spinor fields describe
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s.


Equations of motion

The
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
for the fields are similar to the Hamiltonian equations for discrete particles. For any number of fields: where again the overdots are partial time derivatives, the
variational derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
with respect to the fields :\frac = \frac - \nabla\cdot \frac \,, with · the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
, must be used instead of simply
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s.


Phase space

The fields and conjugates form an infinite dimensional
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
, because fields have an infinite number of degrees of freedom.


Poisson bracket

For two functions which depend on the fields and , their spatial derivatives, and the space and time coordinates, : A = \int d^3 x \mathcal\left(\phi_1,\phi_2,\ldots,\pi_1,\pi_2,\ldots,\nabla\phi_1,\nabla\phi_2,\ldots,\nabla\pi_1,\nabla\pi_2,\ldots,\mathbf,t\right)\,, : B = \int d^3 x \mathcal\left(\phi_1,\phi_2,\ldots,\pi_1,\pi_2,\ldots,\nabla\phi_1,\nabla\phi_2,\ldots,\nabla\pi_1,\nabla\pi_2,\ldots,\mathbf,t\right)\,, and the fields are zero on the boundary of the volume the integrals are taken over, the field theoretic
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
is defined as (not to be confused with the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
from quantum mechanics). : \_ = \int d^3 x \sum_i \left(\frac\frac-\frac\frac\right)\,, where \delta \mathcal/\delta f is the
variational derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
: \frac = \frac - \sum_i \nabla_i \frac \,. Under the same conditions of vanishing fields on the surface, the following result holds for the time evolution of (similarly for ): :\frac = \ + \frac which can be found from the total time derivative of ,
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
, and using the above Poisson bracket.


Explicit time-independence

The following results are true if the Lagrangian and Hamiltonian densities are explicitly time-independent (they can still have implicit time-dependence via the fields and their derivatives),


Kinetic and potential energy densities

The Hamiltonian density is the total energy density, the sum of the kinetic energy density (\mathcal) and the potential energy density (\mathcal), :\mathcal = \mathcal+\mathcal\,.


Continuity equation

Taking the partial time derivative of the definition of the Hamiltonian density above, and using the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
for
implicit differentiation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
and the definition of the conjugate momentum field, gives the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...
: :\frac + \nabla\cdot \mathbf=0 in which the Hamiltonian density can be interpreted as the energy density, and :\mathbf = \frac\frac the energy flux, or flow of energy per unit time per unit surface area.


Relativistic field theory

Covariant Hamiltonian field theory is the relativistic formulation of Hamiltonian field theory. Hamiltonian field theory usually means the symplectic
Hamiltonian formalism In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gene ...
when applied to
classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...
, that takes the form of the instantaneous Hamiltonian formalism on an infinite-dimensional
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
, and where
canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
are field functions at some instant of time. This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
. In Covariant Hamiltonian field theory,
canonical momenta The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical example ...
''pμi'' corresponds to derivatives of fields with respect to all world coordinates ''xμ''. Covariant Hamilton equations are equivalent to the Euler–Lagrange equations in the case of hyperregular Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton–De Donder, polysymplectic, multisymplectic and ''k''-symplecticRey, A., Roman-Roy, N. Saldago, M., Gunther's formalism (''k''-symplectic formalism) in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys. 46 (2005) 052901. variants. A phase space of covariant Hamiltonian field theory is a finite-dimensional polysymplectic or multisymplectic manifold. Hamiltonian non-autonomous mechanics is formulated as covariant Hamiltonian field theory on
fiber bundles In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
over the time axis, i.e. the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
\mathbb.


See also

*
Analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
*
De Donder–Weyl theory In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this frame ...
*
Four-vector In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vect ...
*
Canonical quantization In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible. Historically, this was not quit ...
*
Hamiltonian fluid mechanics Hamiltonian fluid mechanics is the application of Hamiltonian mechanics, Hamiltonian methods to fluid mechanics. Note that this formalism only applies to non-dissipative fluids. Irrotational barotropic flow Take the simple example of a barotropic, ...
*
Covariant classical field theory In mathematical physics, covariant classical field theory represents classical field theory, classical fields by Section (fiber bundle), sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of field ( ...
*
Polysymplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sym ...
*
Non-autonomous mechanics Non-autonomous mechanics describe non- relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space o ...


Notes


Citations


References

* * * *{{cite book , last1=Fetter, first1=A. L., last2=Walecka, first2=J. D., title=Theoretical Mechanics of Particles and Continua, year=1980, isbn= 978-0-486-43261-8, publisher=Dover, pages=258–259 Mathematical physics Classical mechanics Classical field theory Quantum field theory Differential geometry