Covariant Hamiltonian Formulation of Gauge Theories

Outline Outline Covariant Hamiltonian Formulation of Gauge Theories

1 Basics: Action Principle, Gauge Principle, General Relativity J. Struckmeier1,2, D. Vasak3, J. Kirsch3, H. Stöcker1,2,3 2 Relativistic Field Theory in Minkowski Space-Time 1GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany Lagrangian ﬁeld theory [email protected], web-docs.gsi.de/˜struck Covariant Hamiltonian ﬁeld theory 2Fachbereich Physik, Goethe Universität, Frankfurt am Main, Germany Canonical transformations 3Fankfurt Institute for Advanced Studies (FIAS), Frankfurt am Main, Germany 3 Gauge Principle New Horizons in Fundamental Physics: Global symmetry of a dynamical system From Neutron Nuclei via Superheavy Elements and Local symmetry of the amended Klein-Gordon system Supercritical Fields to Neutron Stars and Cosmic Rays Canonical formulation of gauge theory Makutsi Safari Farm, South Africa 4 Extended Formulation of Gauge Theories: Dynamical Space-time 23–28 November 2015 5 Conclusions In honor of the 80th birthday of Walter Greiner

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Basics: Action Principle, Gauge Principle, General Relativity Basics: Action Principle, Gauge Principle, General Relativity

Point Mechanics and Covariant Hamilton Field Theory Basic Principles

Extended Lagrange and Hamilton Formalism for Extended Lagrange and Hamilton Formalism for Point Mechanics and Covariant Hamilton Field Theory Extended Lagrange and Hamilton This book presents the extended Lagrange and Hamilton formalisms 1 of point mechanics and field theory in the usual tensor language of Action Principle: The dynamics of any fundamental theory satisﬁes the standard textbooks on classical dynamics. The notion "extended'' signifies that the physical time of point dynamics as well as the Formalism for Point Mechanics and space-time in field theories are treated as dynamical variables. It action principles. thus elaborates on some important questions including: How do we convert the canonical formalisms of Lagrange and Hamilton that are Covariant Hamilton Field Theory 2 Gauge Principle: Any theory with a global symmetry can be 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 generalized to yield the corresponding locally symmetric theory. General Relativity? 3 The symmetry transformation must be canonical (CT) in order to In this book, the authors demonstrate how the canonical transformation formalism enables us to systematically devise gauge Jürgen Struckmeier theories. With the extended canonical transformation formalism that maintain the action principle. allows to map the space-time geometry, it is possible to formulate Walter Greiner a generalized theory of gauge transformations. For a system that is 4 form-invariant under both a local gauge transformation of the fields The CT rules provide us with the Hamiltonian of the amended, 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. locally form-invariant system. 5 No “model forging” at this fundamental level, but derivation from basic principles. Struckmeier Greiner 6 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. 7 ISBN 978-981-4578-41-7 The change of reference frame must constitute an extended CT, World Scientific www.worldscientific.com World Scientific 9034 hc which also maps the space-time geometry.

3 / 27 4 / 27 Basics: Action Principle, Gauge Principle, General Relativity Basics: Action Principle, Gauge Principle, General Relativity Classical point dynamics Classical ﬁeld dynamics

Lagrange function L Action Principle Euler-Lagrange equations Lagrange density L Action Principle Euler-Lagrange ﬁeld equations ======⇒ ======⇒

w w w w w Legendre transformation w Covariant Legendre transformation

Action Principle Canonical equations of H Hamilton density H Action Principle Canonical ﬁeld equations of H Hamilton function H ======⇒ ======⇒

w w w w w Canonical transformation (CT) w Canonical transformation (CT)

0 0 0 0 Action Principle Canonical equations of H Hamilton density H Action Principle Canonical ﬁeld equations of H Hamilton function H ======⇒ ======⇒

Canonical transformations are those transformations that maintain the A canonical transformation theory in the realm of ﬁeld dynamics emerges form of the action principle and hence the form of the canonical equations. from the covariant (DeDonder-Weyl) Hamiltonian formalism.

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Relativistic Field Theory in Minkowski Space-Time Lagrangian field theory Relativistic Field Theory in Minkowski Space-Time Lagrangian field theory Euler-Lagrange equations in classical field theory Example 1: Klein-Gordon Lagrangian

The dynamics of a system be described by a first-order Lagrangian L that The Lagrangian for a massive charged scalar field is depends on both a scalar field ψ(x) and a 4-vector field aµ(x).

Action principle µ ∂ψ ∂ψ 2 LKG ψ, ψ, ∂ ψ, ∂µψ = α − m ψψ. ∂x ∂xα Z 4 ! ! S = L (ψ, ∂ψ, aµ, ∂aµ, x) d x, δS = 0, δψ ∂R = δaµ ∂R = 0. R The dynamics follow from the Euler-Lagrange field equations δS vanishes for the field evolution that is realized by nature. ∂LKG ∂ψ ∂LKG Calculus of variations: δS = 0 holds exactly if the fields and their partial = , = −m2 ψ, ∂(∂ ψ) ∂x ∂ψ derivatives satisfy the α α hence Euler-Lagrange field equations ∂2ψ ∂2ψ + m2 ψ = 0, + m2 ψ = 0. ∂ ∂L ∂L ∂ ∂L ∂L α α ∂xα∂x ∂xα∂x α − = 0, α − = 0. ∂x ∂(∂αψ) ∂ψ ∂x ∂(∂αaµ) ∂aµ

7 / 27 8 / 27 Relativistic Field Theory in Minkowski Space-Time Lagrangian field theory Relativistic Field Theory in Minkowski Space-Time Covariant Hamiltonian field theory Example: Maxwell Lagrangian DeDonder-Weyl (covariant) Hamiltonian Equivalent description: covariant Hamiltonians A system of vector fields is given by the Maxwell Lagrangian with jµ(x) the 4-current source term Define for the derivatives of each field a 4-vector conjugate momentum field ∂a ∂a µ ∂L νµ ∂L L (a , ∂ a , x) = − 1 f f αβ − jα(x) a , f = ν − µ . π = , p = . M µ ν µ 4 αβ α µν ∂x µ ∂x ν ∂(∂µψ) ∂(∂µaν) µ π : dual quantity of ∂µψ and canonical conjugate of ψ The corresponding Euler-Lagrange equations are νµ p : dual quantity of ∂µaν and canonical conjugate of aν ∂ ∂LM ∂LM The covariant Hamiltonian H is then defined as the complete Legendre − = 0, µ = 0,..., 3. α ∂a ∂x µ ∂aµ transform of the Lagrangian L ∂ ∂x α µ νµ α ∂ψ βα ∂aβ For L we get the particular field equation H(ψ, π , aν, p , x) = π + p − L(ψ, ∂µψ, aν, ∂µaν, x) M ∂x α ∂x α µα 2 α 2 µ ∂f µ ∂ a ∂ a µ The Hesse matrices α + j = 0 ⇔ α − α + j = 0. ! ! ∂x ∂x ∂xµ ∂x ∂xα ∂2L ∂2L , ν ν β This is the tensor form of the inhomogeneous Maxwell equation ∂(∂µψ)∂(∂ ψ) ∂ ∂µaα ∂ ∂ a in Minkowski space. must be non-singular in order for H to exist. 9 / 27 10 / 27

Relativistic Field Theory in Minkowski Space-Time Covariant Hamiltonian field theory Relativistic Field Theory in Minkowski Space-Time Covariant Hamiltonian field theory Example: Maxwell Hamiltonian Covariant canonical field equations

For the Maxwell Lagrangian From the Legendre transformation, we directly obtain the Canonical ﬁeld equations 1 αβ α ∂aν ∂aµ L = − f f − j (x) aα, fµν = − , M 4 αβ ∂x µ ∂x ν ∂H ∂ψ ∂H ∂L ∂πα = , = − = − . µν ν ∂πµ ∂x µ ∂ψ ∂ψ ∂x α the tensor elements p are the dual objects of the derivatives ∂aµ/∂x . να They are obtained from LM via ∂H ∂aν ∂H ∂L ∂p νµ = µ , = − = − α , ∂p ∂x ∂aν ∂aν ∂x µν ∂LM µν µν p = =⇒ p = f , pµν = fµν. ∂ (∂νaµ) They are equivalent to the Euler-Lagrange ﬁeld equations.

The Maxwell Hamiltonian now follows as the Legendre transform This is easily verified for the covariant Hamiltonians µ µ α 2 αβ 1 αβ ∂aα ∂aβ HKG(π , π , ψ, ψ) = παπ + m ψψ HM = p ∂βaα − LM = 2 p − − LM ∂x β ∂x α νµ 1 αβ α HM(aν, p , x) = − 4 p pαβ + j (x) aα. as Remark: only the divergences of the momentum fields are determined by H = − 1 pαβp + jα(x) a , p = −p . M 4 αβ α µν νµ the Hamiltonian and not the individual πµ and pνµ. 11 / 27 12 / 27 Relativistic Field Theory in Minkowski Space-Time Canonical transformations Relativistic Field Theory in Minkowski Space-Time Canonical transformations Requirement of form-invariance for the action principle Canonical transformations in Hamiltonian field theory Provided the dynamics of a system follows from the action principle µ The divergence of a function F1 (ψ, Ψ, aν, Aν, x) is any transformation of the field variables ψ 7→ Ψ, a 7→ A must µ µ α α α α α α maintain the form of the action principle in order to be physical. ∂F1 ∂F1 ∂ψ ∂F1 ∂Ψ ∂F1 ∂aβ ∂F1 ∂Aβ ∂F1 α = α + α + α + α + α The transformations that maintain the form of the action principle are ∂x ∂ψ ∂x ∂Ψ ∂x ∂aβ ∂x ∂Aβ ∂x ∂x expl referred to as canonical. This requirement yields the Comparing the coefficients with the integrand condition yields the Condition for canonical transformations Transformation rules for a generating function F µ Z 1 α ∂ψ βα ∂aβ 4 δ π α + p α − H d x µ µ µ µ R ∂x ∂x ∂F ∂F ∂F ∂F ∂F α πµ = 1 , Πµ = − 1 , pνµ = 1 , Pνµ = − 1 , H0 = H+ 1 Z ∂Ψ ∂A α α βα β 0 4 ∂ψ ∂Ψ ∂aν ∂Aν ∂x expl = δ Π α + P α − H d x. R ∂x ∂x µ The second derivatives of the generating function F1 yield the This condition implies that the integrands may differ by the divergence of µ µ µ symmetry relations for canonical transformations from F1 a vector field F1 (ψ, Ψ, aν, Aν, x) with δF1 |∂R = 0 µ 2 µ µ νµ 2 µ αµ ∂ψ ∂a ∂Ψ ∂A ∂F α ∂π ∂ F1 ∂Π ∂p ∂ F1 ∂P πα + pβα β − H = Πα + Pβα β − H0 + 1 . = = − , = = − ∂x α ∂x α ∂x α ∂x α ∂x α ∂Ψ ∂ψ∂Ψ ∂ψ ∂Aα ∂aν∂Aα ∂aν

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Relativistic Field Theory in Minkowski Space-Time Canonical transformations Gauge Principle µ Generating function of type F2 Gauge Principle: From global to local symmetries

µ We may switch to F2 by means of a Legendre transformation Hamilton density H Global symmetry CT 0 ======⇒ H is form-invariant µ µ νµ µ µ αµ F2 (ψ, Π , aν, P , x) = F1 (ψ, Ψ, aν, Aν, x) + ΨΠ + AαP . w w The set of transformation rules for F µ is then w Render the symmetry transformation local 2 µ Transformation rules for a generating function F 0 2 Hamilton density H Local symmetry CT ======⇒ H is not form-invariant µ µ µ µ ∂F ∂F ∂F ∂F ∂F α w πµ = 2 , Ψδµ = 2 , pνµ = 2 , A δµ = 2 , H0 = H + 2 w ν ν α ν αν α w Add “Gauge Hamiltonian” Hg ∂ψ ∂Π ∂aν ∂P ∂x expl

µ 0 0 Symmetry relations for canonical transformations deﬁned from F2 : Amended H + H Local symmetry CT H + H is form-invariant g ======⇒ g µ 2 µ βµ 2 µ ∂π ∂ F2 µ ∂Ψ ∂p ∂ F2 µ ∂Aα ν = ν = δν , αν = αν = δν ∂Π ∂ψ∂Π ∂ψ ∂P ∂aβ∂P ∂aβ Depending on the particular symmetry group of H, the requirement of form-invariance of H + Hg determines Hg.

15 / 27 16 / 27 Gauge Principle Global symmetry of a dynamical system Gauge Principle Local symmetry of the amended Klein-Gordon system Global phase invariance of the Klein-Gordon system Local phase transformation

The complex Klein-Gordon Lagrangian is form-invariant under the µ To match the derivatives ∂Λ/∂x , a 4-vector gauge ﬁeld aµ must be Global phase transformation incorporated into the theory. The amended theory is form-invariant under the local phase iΛ −iΛ ψ 7→ Ψ = ψ e , ψ 7→ Ψ = ψ e , Λ = const. transformation, if the gauge ﬁeld obeys the transformation rule hence 1 ∂Λ(x) a (x) 7→ A (x) = a (x) + . ∂ψ ∂ψ 2 0 ∂Ψ ∂Ψ 2 µ µ µ g ∂x µ LKG = α − m ψψ, LKG = α − m ΨΨ. ∂x ∂xα ∂x ∂xα

The covariant Klein-Gordon Hamiltonian HKG is form-invariant as well. The transition aµ 7→ Aµ in conjunction with ψ 7→ Ψ maintains the form of the amended Klein-Gordon Lagrangian/Hamiltonian. This is no longer true if the phase transformation is deﬁned to be local, The task is now to work out the amended Klein-Gordon Hamiltonian i.e. explicitly space-time dependent: HKG + Hg to determine the dynamics of the aµ. Local phase transformation This is most transparently worked out by means of the CT formalism.

iΛ(x) −iΛ(x) It is thereby assured that the transformation maintains the action ψ 7→ Ψ = ψ e , ψ 7→ Ψ = ψ e , Λ = Λ(x). principle.

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Gauge Principle Canonical formulation of gauge theory Gauge Principle Canonical formulation of gauge theory

Generating function for the CT of ψ- and gauge fields aµ Transformation rule for the Hamiltonian HKG + Hg µ The generating function F2 for the CT of ψ fields and gauge fields aµ is The transformation of the Hamiltonian H2 = HKG + Hg is determined by µ µ the divergence of the explicitly x -dependent terms of F2 µ µ iΛ(x) µ −iΛ(x) αµ 1 ∂Λ(x) F = Π ψ e + ψ Π e + P aα + 2 α β 2 g ∂x ∂F β ∂Λ 1 ∂ Λ H0 − H = 2 = i Π ψ eiΛ − ψ Πβ e−iΛ + Pαβ . 2 2 ∂x β ∂x β g ∂x α∂x β The subsequent transformation rules for the fields are expl µ µ ∂F ∂F µ Ψδµ = 2 = δµ ψ e−iΛ, πµ = 2 = Π eiΛ The essential point is now to express all Λ-dependencies in terms of old ν ∂Πν ν ∂ψ µ µ and new physical fields ψ, Ψ, aµ, Aµ and their canonical conjugates µ ∂F2 µ iΛ µ ∂F2 µ −iΛ according to the canonical transformation rules. The result is Ψδ = ν = δ ψ e , π = = Π e ν ∂Π ν ∂ψ β ∂A ∂A ∂F µ 1 ∂Λ ∂F µ H0 − H = ig Π Ψ − ΨΠβ A + 1 Pαβ α + β A δµ = 2 = δµ a + , p αµ = 2 = P αµ. 2 2 β 2 ∂x β ∂x α α ν ∂Pαν ν α g ∂x α ∂a α ∂a ∂a − ig πβψ − ψπβ a − 1 pαβ α + β . β 2 ∂x β ∂x α The rules for all fields ψ, ψ, and aα are indeed reproduced. The CT approach simultaneously provides the rules for the conjugate All terms occur symmetrically with opposite sign momentum fields, πµ, πµ, pαµ, and for the Hamiltonian. in the original and the transformed dynamical variables.

19 / 27 20 / 27 Gauge Principle Canonical formulation of gauge theory Gauge Principle Canonical formulation of gauge theory Locally form-invariant Hamiltonian Final locally form-invariant Hamiltonian

Consequently, an amended Hamiltonian H2 of the form To describe dynamical gauge ﬁelds, the locally form-invariant Hamiltonian H2 must be further amended by a term Hdyn for the free gauge ﬁelds α α 1 αβ ∂aα ∂aβ H2 = HKG(π, ψ, x) + ig π ψ − ψπ aα + 2 p β + α ∂x ∂x 1 αβ H3 = H2 + Hdyn, Hdyn = − 4 p pαβ. is transformed according to the genaral rule

α Hdyn is form-invariant according to the canonical transformation rules 0 ∂F2 H2 = H2 + α and thus maintains the form-invariance of H2. ∂x expl 0 The canonical equations following from H3 and H3 are compatible into the new Hamiltonian with the canonical transformation rules.

α ∂A ∂A For a globally form-invariant H, we thus encounter the H0 = H0 (Π, Ψ, x) + ig Π Ψ − ΨΠα A + 1 Pαβ α + β . 2 KG α 2 ∂x β ∂x α Final locally form-invariant Hamiltonian H3 ∂a ∂a As HKG is form-invariant under the global phase transformation, α α 1 αβ 1 αβ α β H3 = HKG(π, ψ, x)+ig π ψ − ψπ aα− 4 p pαβ + 2 p β + α the Hamiltonian H2 = HKG + Hg is form-invariant ∂x ∂x under the corresponding local phase transformation. 21 / 27 22 / 27

Gauge Principle Canonical formulation of gauge theory Gauge Principle Canonical formulation of gauge theory First canonical ﬁeld equation Generalization to a Yang-Mills gauge theory

The canonical formulation of gauge theory can straightforwardly be The ﬁrst canonical equation for H3 is generalized to SU(N), hence to the global symmetry group ∂a ∂H ∂a ∂a µ 3 1 1 µ ν Ψ = U ψ, Ψ = ψ U† ν = µν = − 2 pµν + 2 ν + µ , ∂x ∂p ∂x ∂x ∗ ΨI = uIJ ψJ , ΨI = ψJ uJI hence ∂a ∂a p = ν − µ . µν ∂x µ ∂x ν with the matrix U being unitary in order to preserve the norm ψψ.

pµν is manifestly skew-symmetric in the indices µ, ν. With aKIµ a N × N matrix of real 4-vector gauge fields, the final Hamiltonian emerges as The skew-symmetry of pµν is now derived, not postulated. α α 1 αβ The final locally form-invariant U(1) Hamiltonian then simplifies to H3 = H(π, ψ, x) + ig πK ψJ − ψK πJ aKJα − 4 pJK pKJαβ 1 αβ µν νµ α α 1 αβ µν νµ − 2 ig pJK aKIα aIJβ − aKIβ aIJα , pJK = −pJK . H3 = HKG(π, ψ, x) + ig π ψ − ψπ aα − 4 p pαβ, p = −p . The locally form-invariant SU(N) Hamiltonian exhibits the characteristic self-coupling term, which vanishes in the case of U(1).

23 / 27 24 / 27 Extended Formulation of Gauge Theories: Dynamical Space-time Extended Formulation of Gauge Theories: Dynamical Space-time Extended Gauge Principle: Local Lorentz Transformations Transformation rule for the Hamiltonian of vacuum GR

The generating function is set up to yield the required transformation Hamilton density H Global Lorentz CT H0 is form-invariant η ======⇒ rule for the affine connection coefficients γ αξ . The canonical transformation formalism yields simultaneously the w αξµ w Render the Lorentz transformation local rules for for their conjugates, ˜r , and for the Hamiltonian w η ∂Γη ∂Γη ! 0 0 0 1 ˜ αξµ αξ αµ i η i η Hamilton density H Local Lorentz CT H det Λ − H det Λ = R + − Γ Γ + Γ Γ ======⇒ H is not form-invariant 2 η ∂X µ ∂X ξ αξ iµ αµ iξ ∂γη η ! w 1 αξµ αξ ∂γ αµ i η i η w − ˜r + − γ γ + γ γ w Add “Gauge Hamiltonian” Hg 2 η ∂x µ ∂x ξ αξ iµ αµ iξ The terms emerge in a symmetric form in the original and the 0 0 Amended H + H Local Lorentz CT H + H is form-invariant g ======⇒ g transformed dynamical variables. We encounter the class of Hamiltonians that are form-invariant The requirement of local Lorentz invariance of H + Hg determines Hg, under the transformation of the connection coefficients. hence the dynamics of the space-time geometry. Details can be found in Phys. Rev. D 91, 085030 (2015).

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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 U(1) symmetry group can straightforwardly be generalized to the SU(N) 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.

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