Outline Outline Covariant Hamiltonian Formulation of Gauge Theories J. 1 GSI Struckmeier1,, D. Vasak3, J. Kirsch3, H. 1 Basics:,, General Relativity 3 Global symmetry of a dynamical system Local symmetry of the amended Klein-Gordon system 4 Extended Formulation of Gauge Theories: Dynamical Space-time 5 Conclusions Stöcker1,,3 Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany j.struckmeier@gsi.de, web-docs.gsi.de/ struck Fachbereich 3 Fankfurt Physik, Goethe Universität, Frankfurt am Main, Germany Institute for Advanced Studies FIAS, Frankfurt am Main, Germany New Horizons in Fundamental Physics: From Neutron Nuclei via Superheavy Elements and Supercritical Fields to Neutron Stars and Cosmic Rays Makutsi Safari Farm, South Africa 3 8 November 015 In honor of the 80th birthday of Walter Greiner 1 / 7 Basics:,, General Relativity This book presents the extended Lagrange and Hamilton formalisms of point mechanics and field theory in the usual tensor language of standard textbooks on classical dynamics. The notion "extended'' signifies that the physical time of point dynamics as well as the space-time in field theories are treated as dynamical variables. It thus elaborates on some important questions including: How do we convert the canonical formalisms of Lagrange and Hamilton that are built upon Newton's concept of an absolute time into the appropriate form of the post-einstein era? How do we devise a Hamiltonian field theory with space-time as a dynamical variable in order to also cover General Relativity? In this book, the authors demonstrate how the canonical transformation formalism enables us to systematically devise gauge theories. With the extended canonical transformation formalism that allows to map the space-time geometry, it is possible to formulate a generalized theory of gauge transformations. For a system that is form-invariant under both a local gauge transformation of the fields and under local variations of the space-time geometry, we will find a formulation of General Relativity to emerge naturally from basic principles rather than being postulated. Basics:,, General Relativity Extended Lagrange and Hamilton Formalism for Point Mechanics and Covariant Hamilton Field Theory Extended Lagrange and Hamilton Formalism for Point Mechanics and Covariant Hamilton Field Theory / 7 Basic Principles Extended Lagrange and Hamilton Formalism for Point Mechanics and Covariant Hamilton Field Theory 1 3 Jürgen Struckmeier Walter Greiner 4 5 Struckmeier Greiner World Scientific www.worldscientific.com 9034 hc 6 7 ISBN 978-981-4578-41-7 World Scientific 3 / 7 : The dynamics of any fundamental theory satisfies the action principles. : Any theory with a global symmetry can be generalized to yield the corresponding locally symmetric theory. The symmetry transformation must be canonical CT in order to maintain the action principle. The CT rules provide us with the Hamiltonian of the amended, locally form-invariant system. No model forging at this fundamental level, but derivation from basic principles. General Principle of Relativity: The form of the action principle and hence the resulting field equations should be the same in any frame of reference. The change of reference frame must constitute an extended CT, which also maps the space-time geometry. 4 / 7
Basics:,, General Relativity Classical point dynamics Basics:,, General Relativity Classical field dynamics Lagrange function L Legendre transformation Euler-Lagrange equations Lagrange density L Covariant Legendre transformation Euler-Lagrange field equations Hamilton function H Canonical equations of H Canonical field equations of H Canonical transformation CT Canonical transformation CT Hamilton function H Canonical equations of H Canonical field equations of H are those transformations that maintain the form of the action principle and hence the form of the canonical equations. 5 / 7 A canonical transformation theory in the realm of field dynamics emerges from the covariant DeDonder-Weyl Hamiltonian formalism. 6 / 7 Euler-Lagrange equations in classical field theory The dynamics of a system be described by a first-order Lagrangian L that depends on both a scalar field ψx and a 4-vector field a µ x. Action principle S = R L ψ, ψ, a µ, a µ, x d 4 x, δs! = 0, δψ R = δa µ R! = 0. δs vanishes for the field evolution that is realized by nature. Calculus of variations: δs = 0 holds exactly if the fields and their partial derivatives satisfy the Euler-Lagrange field equations L ψ L ψ = 0, L a µ L = 0. a µ Example 1: Klein-Gordon Lagrangian The Lagrangian for a massive charged scalar field is L KG ψ, ψ, µ ψ, µ ψ = ψ ψ m ψψ. The dynamics follow from the Euler-Lagrange field equations hence L KG ψ = ψ, ψ + m ψ = 0, L KG ψ = m ψ, ψ + m ψ = 0. 7 / 7 8 / 7
Example: Maxwell Lagrangian A system of vector fields is given by the Maxwell Lagrangian with j µ x the 4-current source term L M a µ, ν a µ, x = 1 4 f f j x a, The corresponding Euler-Lagrange equations are f µν = a ν x µ a µ x ν. L M aµ L M = 0, µ = 0,..., 3. a µ For L M we get the particular field equation f µ + jµ = 0 a x µ a µ + j µ = 0. This is the tensor form of the inhomogeneous Maxwell equation in Minkowski space. 9 / 7 DeDonder-Weyl covariant Hamiltonian Equivalent description: covariant Hamiltonians Define for the derivatives of each field a 4-vector conjugate momentum field π µ = L µ ψ, pνµ = L µ a ν. π µ : dual quantity of µ ψ and canonical conjugate of ψ p νµ : dual quantity of µ a ν and canonical conjugate of a ν The covariant Hamiltonian H is then defined as the complete Legendre transform of the Lagrangian L Hψ, π µ, a ν, p νµ, x = π ψ + p a Lψ, µψ, a ν, µ a ν, x The Hesse matrices L µ ψ ν, ψ must be non-singular in order for H to exist. L µ a ν a 10 / 7 Example: Maxwell Hamiltonian For the Maxwell Lagrangian L M = 1 4 f f j x a, f µν = a ν x µ a µ x ν, the tensor elements p µν are the dual objects of the derivatives a µ / x ν. They are obtained from L M via p µν = L M ν a µ = p µν = f µν, p µν = f µν. Covariant canonical field equations From the Legendre transformation, we directly obtain the Canonical field equations π µ = ψ x µ, p νµ = a ν x µ, ψ = L ψ = π. = L = pν a ν a ν, They are equivalent to the Euler-Lagrange field equations. The Maxwell Hamiltonian now follows as the Legendre transform H M = p a L M = 1 a p x a L M as H M = 1 4 p p + j x a, p µν = p νµ. 11 / 7 This is easily verified for the covariant Hamiltonians H KG π µ, π µ, ψ, ψ = π π + m ψψ H M a ν, p νµ, x = 1 4 p p + j x a. Remark: only the divergences of the momentum fields are determined by the Hamiltonian and not the individual π µ and p νµ. 1 / 7
Requirement of form-invariance for the action principle Provided the dynamics of a system follows from the action principle any transformation of the field variables ψ Ψ, a µ A µ must maintain the form of the action principle in order to be physical. The transformations that maintain the form of the action principle are referred to as canonical. This requirement yields the Condition for canonical transformations [ δ π ψ R + a ] p H d 4 x [ = δ Π Ψ + A ] P H d 4 x. R This condition implies that the integrands may differ by the divergence of a vector field F µ 1 ψ, Ψ, a ν, A ν, x with δf µ 1 R = 0 π ψ + a p Ψ H = Π + A P H + F 1. 13 / 7 in Hamiltonian field theory The divergence of a function F µ 1 ψ, Ψ, a ν, A ν, x is F1 = F 1 ψ ψ + F 1 Ψ Ψ + F 1 a a + F 1 A A + F 1 Comparing the coefficients with the integrand condition yields the Transformation rules for a generating function F µ 1 π µ = F µ 1 ψ, Πµ = F µ 1 Ψ, pνµ = F µ 1, P νµ = F µ 1, H = H+ F 1 a ν A ν The second derivatives of the generating function F µ 1 yield the symmetry relations for canonical transformations from F µ 1 π µ Ψ = F µ 1 ψ Ψ = Πµ ψ, p νµ A = F µ 1 a ν A = Pµ a ν 14 / 7 Generating function of type F µ : From global to local symmetries We may switch to F µ by means of a Legendre transformation F µ ψ, Πµ, a ν, P νµ, x = F µ 1 ψ, Ψ, a ν, A ν, x + ΨΠ µ + A P µ. The set of transformation rules for F µ is then Transformation rules for a generating function F µ π µ = F µ ψ, Ψδµ ν = F µ Π ν, pνµ = F µ, A δ ν µ = F µ a ν P ν, H = H + F Global symmetry CT == H is form-invariant Render the symmetry transformation local Local symmetry CT == H is not form-invariant Add Gauge Hamiltonian Hg Symmetry relations for canonical transformations defined from F µ : π µ Π ν = F µ ψ Π ν = Ψ δµ ν ψ, p µ P ν = F µ a P ν = A δµ ν a Amended H + H g Local symmetry CT == H + H g is form-invariant Depending on the particular symmetry group of H, the requirement of form-invariance of H + H g determines H g. 15 / 7 16 / 7
Global symmetry of a dynamical system Local symmetry of the amended Klein-Gordon system Global phase invariance of the Klein-Gordon system The complex Klein-Gordon Lagrangian is form-invariant under the Global phase transformation hence ψ Ψ = ψ e iλ, ψ Ψ = ψ e iλ, Λ = const. L KG = ψ ψ m ψψ, L KG = Ψ Ψ m ΨΨ. The covariant Klein-Gordon Hamiltonian H KG is form-invariant as well. This is no longer true if the phase transformation is defined to be local, i.e. explicitly space-time dependent: Local phase transformation ψ Ψ = ψ e iλx, ψ Ψ = ψ e iλx, Λ = Λx. 17 / 7 Local phase transformation To match the derivatives Λ/ x µ, a 4-vector gauge field a µ must be incorporated into the theory. The amended theory is form-invariant under the local phase transformation, if the gauge field obeys the transformation rule a µ x A µ x = a µ x + 1 g Λx x µ. The transition a µ A µ in conjunction with ψ Ψ maintains the form of the amended Klein-Gordon Lagrangian/Hamiltonian. The task is now to work out the amended Klein-Gordon Hamiltonian H KG + H g to determine the dynamics of the a µ. This is most transparently worked out by means of the CT formalism. It is thereby assured that the transformation maintains the action principle. 18 / 7 Generating function for the CT of ψ- and gauge fields a µ The generating function F µ for the CT of ψ fields and gauge fields a µ is F µ = Πµ ψ e iλx + ψ Π µ e iλx + P µ a + 1 g The subsequent transformation rules for the fields are Λx Ψδ µ ν = F µ Π ν = δµ ν ψ e iλ, π µ = F µ ψ = Πµ e iλ Ψδ ν µ = F µ Π ν = δ ν µ ψ e iλ, π µ = F µ ψ = Πµ e iλ A δ ν µ = F µ P ν = δµ ν a + 1 Λ g, p µ = F µ = P µ. a The rules for all fields ψ, ψ, and a are indeed reproduced. The CT approach simultaneously provides the rules for the conjugate momentum fields, π µ, π µ, p µ, and for the Hamiltonian. 19 / 7 Transformation rule for the Hamiltonian H KG + H g The transformation of the Hamiltonian H = H KG + H g is determined by the divergence of the explicitly x µ -dependent terms of F µ H H = F x = i Π ψ e iλ ψ Π e iλ Λ x + 1 Λ g P x. The essential point is now to express all Λ-dependencies in terms of old and new physical fields ψ, Ψ, a µ, Aµ and their canonical conjugates according to the canonical transformation rules. The result is H H = ig Π Ψ ΨΠ A + 1 A P x + A ig π ψ ψπ a 1 a p x + a. All terms occur symmetrically with opposite sign in the original and the transformed dynamical variables. 0 / 7
Locally form-invariant Hamiltonian Consequently, an amended Hamiltonian H of the form H = H KG π, ψ, x + ig is transformed according to the genaral rule H = H + F into the new Hamiltonian H = H KGΠ, Ψ, x + ig π ψ ψπ a + 1 a p x + a Π Ψ ΨΠ A + 1 A P x + A. As H KG is form-invariant under the global phase transformation, the Hamiltonian H = H KG + H g is form-invariant under the corresponding local phase transformation. 1 / 7 Final locally form-invariant Hamiltonian To describe dynamical gauge fields, the locally form-invariant Hamiltonian H must be further amended by a term H dyn for the free gauge fields H 3 = H + H dyn, H dyn = 1 4 p p. H dyn is form-invariant according to the canonical transformation rules and thus maintains the form-invariance of H. The canonical equations following from H 3 and H 3 are compatible with the canonical transformation rules. For a globally form-invariant H, we thus encounter the Final locally form-invariant Hamiltonian H 3 H 3 = H KG π, ψ, x+ig π ψ ψπ a 1 4 p p + 1 a p x + a / 7 First canonical field equation The first canonical equation for H 3 is hence a µ x ν = 3 p µν = 1 p µν + 1 aµ x ν + a ν x µ, p µν = a ν x µ a µ x ν. p µν is manifestly skew-symmetric in the indices µ, ν. The skew-symmetry of p µν is now derived, not postulated. The final locally form-invariant U1 Hamiltonian then simplifies to H 3 = H KG π, ψ, x + ig π ψ ψπ a 1 4 p p, p µν = p νµ. 3 / 7 Generalization to a Yang-Mills gauge theory The canonical formulation of gauge theory can straightforwardly be generalized to SUN, hence to the global symmetry group Ψ = U ψ, Ψ = ψ U Ψ I = u IJ ψ J, Ψ I = ψ J u JI with the matrix U being unitary in order to preserve the norm ψψ. With a KIµ a N N matrix of real 4-vector gauge fields, the final Hamiltonian emerges as H 3 = Hπ, ψ, x + ig π K ψ J ψ K πj a KJ 1 4 p JK p KJ a KI a IJ a KI a IJ, p µν JK = pνµ JK. 1 ig p JK The locally form-invariant SUN Hamiltonian exhibits the characteristic self-coupling term, which vanishes in the case of U1. 4 / 7
Extended Formulation of Gauge Theories: Dynamical Space-time Extended : Local Lorentz Transformations Amended H + H g Global Lorentz CT = H is form-invariant Render the Lorentz transformation local Local Lorentz CT = H is not form-invariant Add Gauge Hamiltonian Hg Local Lorentz CT = H + H g is form-invariant The requirement of local Lorentz invariance of H + H g determines H g, hence the dynamics of the space-time geometry. 5 / 7 Extended Formulation of Gauge Theories: Dynamical Space-time Transformation rule for the Hamiltonian of vacuum GR The generating function is set up to yield the required transformation rule for the affine connection coefficients γ η ξ. The canonical transformation formalism yields simultaneously the rules for for their conjugates, r η ξµ, and for the Hamiltonian H det Λ H det Λ = 1 ξµ Γ η ξ R η Γ i ξγ η iµ + Γi µγ η iξ 1 r ξµ η X µ + Γη µ X ξ γ η ξ x µ + γη µ x ξ γ i ξ γ η iµ + γi µγ η iξ The terms emerge in a symmetric form in the original and the transformed dynamical variables. We encounter the class of Hamiltonians that are form-invariant under the transformation of the connection coefficients. Details can be found in Phys. Rev. D 91, 085030 015. 6 / 7 Conclusions Conclusions Gauge theories are most naturally formulated as canonical transformations. This automatically ensures the action principle to be maintained. Then no additional structure such as the minimum coupling rule needs to be incorporated into the derivation. Remarkably, all we need to know is the given system s global symmetry group in order to work out the local symmetry group, hence the corresponding locally form-invariant system. The theory presented here for the U1 symmetry group can straightforwardly be generalized to the SUN symmetry group. The theory can be further generalized to accommodate General Relativity. The global symmetry group is then given by the Lorentz group. The requirement of the system s form-invariance under local Lorentz transformations then provides a description of the space-time dynamics. 7 / 7