U(1)-gauge theory via canonical transformations Adrian Knigstein - - PowerPoint PPT Presentation

U(1)-gauge theory via canonical transformations

Adrian Königstein

Institut für Theoretische Physik, Johann Wolgang Goethe-Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main

18 January 2016

in coorperation with: Johannes Kirsch, Horst Stöcker, Jürgen Struckmeier, David Vasak, Matthias Hanauske

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 1 / 45

Overview

1

Revision of U(1)-gauge theory

2

Revision of Hamilton mechanics and canonical transformations

3

Covariant Hamilton field theory

4

Covariant canonical transformations

5

Scalar electrodynamics via canonical transformations Local phase transformations as canonical transformations Gauge field dynamics

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 2 / 45

Revision of U(1)-gauge theory

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 3 / 45

Revision of U(1)-gauge theory

Complex Klein-Gordon Lagrange density

L = (∂µ ¯ φ)(∂µφ) − m2 ¯ φφ . The complex Klein Gordon Lagrange density is invariant under global U(1)-transformations. φ − → Φ = e−iΛ φ , ¯ φ − → ¯ Φ = ¯ φ eiΛ .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 4 / 45

Revision of U(1)-gauge theory

Füge Bild Rischke ein mit, mit lokaler Phasentransformation. Phase transformations should be local, not only global! φ − → Φ = e−iΛ(x) φ , ¯ φ − → ¯ Φ = ¯ φ eiΛ(x) , L(Φ, ¯ Φ, ∂αΦ, ∂α ¯ Φ, x) ! = L(φ, ¯ φ, ∂αφ, ∂α ¯ φ, x) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 5 / 45

Revision of U(1)-gauge theory

Problem: The complex Klein-Gordon Lagrangian is not invariant under local U(1)-transformations. L(Φ, ¯ Φ, ∂αΦ, ∂α ¯ Φ, x) = L(φ, ¯ φ, ∂αφ, ∂α ¯ φ, x) + i

¯

φ(∂µφ) − (∂µ ¯ φ)φ

• (∂µΛ) + ¯

φ φ (∂µΛ)(∂µΛ) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 6 / 45

Revision of U(1)-gauge theory

Problem: The complex Klein-Gordon Lagrangian is not invariant under local U(1)-transformations. L(Φ, ¯ Φ, ∂αΦ, ∂α ¯ Φ, x) = L(φ, ¯ φ, ∂αφ, ∂α ¯ φ, x) + i

¯

φ(∂µφ) − (∂µ ¯ φ)φ

• (∂µΛ) + ¯

φ φ (∂µΛ)(∂µΛ) . Solution: Introduce compensatory fields with a specific transformation behaviour. ∂µ − → Dµ ≡ ∂µ − iq aµ , Aµ = aµ + 1 q (∂µΛ) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 6 / 45

Revision of U(1)-gauge theory

The modified theory (scalar electrodynamics) reads L(φ, ¯ φ, ∂βφ, ∂β ¯ φ, aα, ∂βaα) = (D∗

µ ¯

φ)(Dµφ) − m2 ¯ φ φ + LA . With an additional dynamical/kinetical term for the gauge fields, e.g. LA = −1 4FµνF µν .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 7 / 45

What is my goal?

• 1. U(1)-transformations via canonical transformations

In point mechanics transformations that leave the action invariant are formulated as canonical transformations. Question: Is there a possibility to formulate local phase transformations as canonical transformations?

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 8 / 45

What is my goal?

• 1. U(1)-transformations via canonical transformations

In point mechanics transformations that leave the action invariant are formulated as canonical transformations. Question: Is there a possibility to formulate local phase transformations as canonical transformations?

• 2. Electrodynamics as an manifestly covariant Hamilton field theory

In point mechanics canonical transformations are powerful method in Hamilton mechanics, whereas in common field theory this tool is missing. Question: Does a covariant Hamilton field theory with canonical transformations actually exist and is it utile in the context of U(1)-transformations?

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 8 / 45

Revision of Hamilton mechanics and canonical transformations

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 9 / 45

Revision of Hamilton mechanics and canonical transformations

The Hamilton function is defined as H(q, p, t) = p ˙ q − L(q, ˙ q, t) with corresponding equations of motion (canonical equations) ∂H ∂q = −˙ p ∂H ∂p = +˙ q

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 10 / 45

Revision of Hamilton mechanics and canonical transformations

Lagrange funktions are not unique L(q, ˙ q, t) = L(Q, ˙ Q, t) + ˙ f (q, Q, t) , which induces the definition of a canonical transformation p ˙ q − H(q, p, t) = P ˙ Q − H(Q, P, t) + ˙ f (q, Q, t) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 11 / 45

Revision of Hamilton mechanics and canonical transformations

The generating function f1(q, Q, t): p = ∂f1 ∂q , P = −∂f1 ∂Q , H(Q, P, t) = H(q, p, t) + ∂f1 ∂t . The corresponding symmetry relation is ∂p ∂Q = ∂2f1 ∂q ∂Q = −∂P ∂q .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 12 / 45

Revision of Hamilton mechanics and canonical transformations

In total there are four generating functions f1(q, Q, t), f2(q, P, t), f3(p, Q, t) and f4(p, P, t), which are connected by Legendre transformation, e.g. f2(q, P, t) = P Q + f1(q, Q, t) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 13 / 45

Revision of Hamilton mechanics and canonical transformations

Q P q f1(q, Q, t) f2(q, P, t) p = ∂f1

∂q , P = − ∂f1 ∂Q

p = ∂f2

∂q , Q = ∂f2 ∂P

p f3(p, Q, t) f4(p, P, t) q = −∂f3

∂p , P = − ∂f3 ∂Q

q = −∂f4

∂p , Q = ∂f4 ∂P

Table: Generating functions and transformation laws for canonical transformations

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 14 / 45

Revision of Hamilton mechanics and canonical transformations

Example: Harmonic Oscillator H = 1 2 mp2 + m ω2 2 q2 . We choose f1(q, Q) = m ω0 2 q2 cot(Q) , which yields p = ∂f1 ∂q = m ω0q cot(Q) , P = −∂f1 ∂Q = m ω0 2 q2 1 sin(Q) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 15 / 45

Revision of Hamilton mechanics and canonical transformations

Therefore the transformation laws are q =

• 2 P

m ω0 sin(Q) , p =

• 2 P m ω0 cos(Q)

and the transformed Hamilton function reads H(Q, P, t) = ω0P

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 16 / 45

Covariant Hamilton field theory

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 17 / 45

Covariant Hamilton formalism

In conventional field theory the Hamilton density is defined as H(φ, π, ∇φ, x) ≡ π ˙ φ − L(φ, ∂αφ, x) , where π ≡ ∂L ∂(∂0φ) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 18 / 45

Covariant Hamilton field theory

In a manifestly covariant Hamilton field theory, also called DeDonder-Weyl theory, the canonically conjugate field is defined by πµ ≡ ∂L ∂(∂µφ) which induces a covariant Legendre transformation for the Hamilton density H(φ, πα, x) ≡ πµ∂µφ − L(φ, ∂αφ, x).

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 19 / 45

Covariant Hamilton field theory

Corresponding canonical equations are ∂H ∂φ = −∂µπµ , ∂H ∂πµ = ∂µφ , ∂µH|expl = − ∂µL|expl .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 20 / 45

Covariant Hamilton field theory

Example: The real Klein-Gordon Hamiltonian L = 1 2

• (∂µφ)(∂µφ) − m2φ2

. Consequently the canonically conjugate field is πµ = ∂L ∂(∂µφ) = (∂µφ) and we find the following expression for the Hamilton density H(φ, πα, x) = πµ∂µφ − L(φ, ∂αφ, x) = πµ∂µφ − 1 2

• (∂µφ)(∂µφ) − m2φ2

= 1 2

• πµ πµ + m2φ2

.

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 21 / 45

Covariant Hamilton field theory

Calculating the canonical equations −∂µπµ = ∂H ∂φ = m2φ , ∂µφ = ∂H ∂πµ = πµ . Inserting one into the other we find the Klein-Gordon equation ( + m2)φ = 0 .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 22 / 45

Covariant canonical transformations

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 23 / 45

Covariant canonical transformations

Covariant canonical transformations are defined in complete analogy to point mechanics πµ∂µφ − H(φ, πα, x) = Πµ∂µΦ − ˜ H(Φ, Πβ, x) + ∂µf µ

1 (φ, Φ, x) .

Φ Πµ φ f µ

1 (φ, Φ, x)

f µ

2 (φ, Πβ x)

πµ = ∂f µ

1

∂φ , Πµ = −∂f µ

1

∂Φ

πµ = ∂f µ

2

∂φ , δµ ν Φ = ∂f µ

2

∂Πν

πµ f µ

3 (πα, Φ, x)

f µ

4 (πα, Πβ, x)

δµ

ν φ = − ∂f µ

3

∂πν , Πµ = −∂f µ

3

∂Φ

δµ

ν φ = − ∂f µ

4

∂πν , δµ ν Φ = ∂f µ

4

∂Πν

Table: Generating functions and transformation laws for covariant canonical transformations

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 24 / 45

Covariant canonical transformations

For each generating function a symmetry relation can be derived ∂πµ ∂Φ = ∂2f µ

1

∂φ ∂Φ = −∂Πµ ∂φ , ∂πµ ∂Πν = ∂2f µ

2

∂φ ∂Πν = δµ

ν

∂Φ ∂φ , δµ

ν

∂φ ∂Φ = − ∂2f µ

3

∂πν ∂Φ = ∂Πµ ∂πν , δµ

α

∂φ ∂Πβ = − ∂2f µ

4

∂πα ∂Πβ = −δµ

β

∂Φ ∂πα . The transformation law for the Hamilton densities is equivalent for all generating functions H′ = H + ∂µf µ

• |expl .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 25 / 45

Scalar electrodynamics via canonical transformations

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 26 / 45

Local phase transformations as canonical transformations

The starting point for the further discussion is a Hamilton density H(φ, ¯ φ, ¯ πα, πα, x) = ¯ πµπµ + V (φ, ¯ φ, x) , which is assumed to be invariant under global phase transformations φ − → Φ = e−iΛ φ , ¯ φ − → ¯ Φ = ¯ φ eiΛ .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 27 / 45

Local phase transformations as canonical transformations

In fact we want our system to be invariant under local phase transformations φ − → Φ = e−iΛ(x) φ , ¯ φ − → ¯ Φ = ¯ φ eiΛ(x) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 28 / 45

Local phase transformations as canonical transformations

In fact we want our system to be invariant under local phase transformations δµ

ν Φ = ∂f µ 2

∂ ¯ Πν

!

= δµ

ν e−iΛ φ ,

δµ

ν ¯

Φ = ∂f µ

2

∂Πν

!

= δµ

ν ¯

φ eiΛ .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 29 / 45

Local phase transformations as canonical transformations

In fact we want our system to be invariant under local phase transformations δµ

ν Φ = ∂f µ 2

∂ ¯ Πν

!

= δµ

ν e−iΛ φ ,

δµ

ν ¯

Φ = ∂f µ

2

∂Πν

!

= δµ

ν ¯

φ eiΛ . Functional integration yields f µ

2 = ¯

Πµe−iΛ φ + ¯ φ eiΛΠµ + cµ(φ, ¯ φ, x) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 29 / 45

Local phase transformations as canonical transformations

We use the transformation rules of the generating function f µ

2 to derive

the transformation laws of the canonically conjugate fields πµ = ∂f µ

2

∂ ¯ φ = eiΛΠµ + ∂cµ ∂ ¯ φ , ¯ πµ = ∂f µ

2

∂φ = ¯ Πµe−iΛ + ∂cµ ∂φ .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 30 / 45

Local phase transformations as canonical transformations

πµ = ∂f µ

2

∂ ¯ φ = eiΛΠµ , ¯ πµ = ∂f µ

2

∂φ = ¯ Πµe−iΛ . The restrictions of the symmetry relations lead to the first modification of the generating function. f µ

2 = ¯

Πµe−iΛ φ + ¯ φ eiΛΠµ + wµ(x) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 31 / 45

Local phase transformations as canonical transformations

f µ

2 = ¯

Πµe−iΛ φ + ¯ φ eiΛΠµ + wµ(x) . The most interesting and most important transformation behaviour is the

• ne of the Hamilton density, which is

H′ − H = −i(∂µΛ)

• ¯

πµ φ − ¯ φ πµ + ∂µwµ(x) . The Hamilton density is not form-invariant!

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 32 / 45

Local phase transformations as canonical transformations

To compensate the non-vanishing term, an extra gauge field is introduced to improve the original Hamiltonian. Ha ≡ H + iq

• ¯

πµ φ − ¯ φ πµ aµ . This new Hamiltonian is said to be form-invariant, which means that H′

a = H′ + iq

¯

Πµ Φ − ¯ Φ Πµ Aµ − ∂µwµ and 0 = H′

a − Ha .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 33 / 45

Local phase transformations as canonical transformations

The intermediate results are Ha(φ, ¯ φ, ¯ πα, πα, aα, pαγ, x) ≡ H(φ, ¯ φ, ¯ πα, πα, x) + iq

• ¯

πµ φ − ¯ φ πµ aµ and Aµ = aµ + 1 q (∂µΛ) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 34 / 45

Gauge field dynamics

The transformation behaviour of the gauge field has to be included in the generating function f µ

2 . We claim

δµ

ν Aα = ∂f µ 2

∂Pαν

!

= δµ

ν

• aα + 1

q (∂αΛ)

• .

Functional integration yields f µ

2 = ¯

Πµe−iΛ φ + ¯ φ eiΛΠµ + Pαµ

• aα + 1

q (∂αΛ)

• + c′µ(aα, x) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 35 / 45

Gauge field dynamics

The transformation behaviour of the canonically conjugate field is then pνµ = ∂f µ

2

∂aν = Pνµ + ∂c′µ ∂aν .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 36 / 45

Gauge field dynamics

The transformation behaviour of the canonically conjugate field is then pνµ = ∂f µ

2

∂aν = Pνµ + ∂c′µ ∂aν . Symmetry relations restrict the integration constant. c′µ(aα, x) = yνµ(x) aν + wµ(x)

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 36 / 45

Gauge field dynamics

The transformation behaviour of the canonically conjugate field is then pνµ = ∂f µ

2

∂aν = Pνµ + yνµ . Symmetry relations restrict the integration constant. c′µ(aα, x) = yνµ(x) aν + wµ(x) .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 37 / 45

Gauge field dynamics

The generating function is modified f µ

2 = ¯

Πµe−iΛ φ + ¯ φ eiΛΠµ + Pαµ

• aα + 1

q (∂αΛ)

• + yνµ aν + wµ

which leads to a modified transformation law for the Hamilton density H′ − H = ∂µf µ

2 |expl

= −i(∂µΛ)

¯

Πµe−iΛ φ − ¯ φ eiΛΠµ + 1 q Pµν (∂µ∂νΛ) + (∂µyνµ) aν + ∂µwµ .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 38 / 45

Gauge field dynamics

We choose the integration constants as follows yνµ = 1 q [−ηνµ(Λ) + (∂ν∂µΛ)] , wµ = 1 2 q (∂νΛ) yνµ + w′µ .

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 39 / 45

Gauge field dynamics

Using all previous transformation laws we find H′ + iq

¯

ΠµΦ − ¯ Φ Πµ Aµ + 1 2PβαPβα − 1 6P2 = H + iq

• ¯

πµφ − ¯ φ πµ aµ + 1 2pβαpβα − 1 6p2 + ∂µw′µ . Therefore the final Hamiltonian is given by Ha(φ, ¯ φ, ¯ πα, πα, aα, pαγ, x) = H + iq

• ¯

πµφ − ¯ φ πµ aµ + 1 2pβαpβα − 1 6p2 . A Legendre transformation provides the well known Klein-Gordon-Maxwell system - scalar electrodynamics.

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 40 / 45

References I

• T. Cheng and L. Li, “Gauge theory of elementary particle physics”,

Clarendon press Oxford, (1984).

• T. De Donder, “Théorie Invariantive Du Calcul des Variations”, Paris,

Gaulthier-Villars & Cie,(1930).

• H. Dehnen and J. Petry, “Theorie der Elementarteilchen und ihrer

Wechselwirkungen”, Univ.,Fak. für Physik, (1999).

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Bewegungsgleichungen für einen geladenen Massepunkt”, Zeitschrift für Physik 39:226-232, (1926).

• S. Gasiorowicz, “Elementary particle physics”, New York, Wiley,

(1966).

• J. Good, “Hamilton Mechanics of Fields”, Phys. Rev. 93(1):239-243,

(1954).

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 41 / 45

References II

• W. Greiner, “Classical mechanics, 2nd ed.”, Berlin, Germany: Springer,

(2010).

• W. Greiner and J. Reinhardt, “Field quantization”, Berlin, Germany:

Springer (1996).

• C. Günther, “The polysymplectic Hamiltonian formalism in field theory

and calculus of variations I: The local case”, J. Differential Geometry 25:23, (1987).

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QCD-motivierten Modellen”, PhD thesis, Univ. Frankfurt (Main), (2005).

• J. V. José and E. J. Saletan, “Classical Dynamics”, Cambridge,

Cambridge University Press, (1998).

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 42 / 45

References III

• A. Koenigstein, F. Giacosa and D. H. Rischke, “Classical and quantum

theory of the massive spin-two field”, arXiv:1508.00110 [hep-th], (2015).

• D. Musicki, “On canonical formalism in field theory with derivatives of

higher order - canonical transformations”, J. Phys A:Math. Gen. 11:39, (1978).

• P. C. Paufler, “Multisymplektische Feldtheorie”, PhD thesis, University
• f Freiburg im Breisgau, Germany, (2001).
• G. Sardanashvily, “Generalized Hamiltonian Formalism for Field

Theory”, Singapore, World Scientific Publishing Co., (1995).

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 43 / 45

References IV

• J. Struckmeier and H. Reichau, “Generalized U(N) gauge

transformations in the realm of the extended covariant Hamilton formalism of field theory”, J. Phys. G: Nucl. Part. Phys. 40:015007, (2013).

• J. Struckmeier and A. Redelbach, “Covariant hamiltonian field

theory”, Int. J. Mod. Phys. E 17:435-491, (2008).

• G. t’Hooft, “Under the Spell of the Gauge Principle”, Singapore,

World Scientific Publishing Co., (1994).

• V. Tapia, “Covariant Field Theory and Surface Terms”, Il Nuovo

Cimento 102 B(2):123-130, (1988).

• S. Weinberg, “The Quantum Theory of Fields, volume I”, Cambridge

University Press, (1996).

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 44 / 45

References V

• H. Weyl, “Elektron und Gravitation”, Zeitschrift für Physik

56:330-352, (1929).

• H. Weyl, “Geodesic fields in the calculus of variation for multiple

integrals”, Annals of Mathematics 36:607, (1935).

Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 45 / 45