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 Revision of U(1)-gauge theory 1 Revision of Hamilton mechanics and canonical transformations 2 Covariant Hamilton field theory 3 Covariant canonical transformations 4 Scalar electrodynamics via canonical transformations 5 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 φ )( ∂ µ φ ) − m 2 ¯ L = ( ∂ µ ¯ φφ . The complex Klein Gordon Lagrange density is invariant under global U (1)-transformations. → Φ = e − i Λ φ , φ − φ e i Λ . ¯ → ¯ Φ = ¯ φ − 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 = e − i Λ( x ) φ , φ − ¯ → ¯ Φ = ¯ φ e i Λ( x ) , φ − L (Φ , ¯ Φ , ∂ α Φ , ∂ α ¯ Φ , x ) ! = L ( φ, ¯ φ, ∂ α φ, ∂ 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 , ∂ α Φ , ∂ α ¯ Φ , x ) = L ( φ, ¯ φ, ∂ α φ, ∂ α ¯ φ, x ) φ ( ∂ µ φ ) − ( ∂ µ ¯ � ¯ � ( ∂ µ Λ) + ¯ φ φ ( ∂ µ Λ)( ∂ µ Λ) . + i φ ) φ 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 , ∂ α Φ , ∂ α ¯ Φ , x ) = L ( φ, ¯ φ, ∂ α φ, ∂ α ¯ φ, x ) φ ( ∂ µ φ ) − ( ∂ µ ¯ � ¯ � ( ∂ µ Λ) + ¯ φ φ ( ∂ µ Λ)( ∂ µ Λ) . + i φ ) φ Solution: Introduce compensatory fields with a specific transformation behaviour. ∂ µ − → D µ ≡ ∂ µ − iq a µ , A µ = a µ + 1 q ( ∂ µ Λ) . 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 φ )( D µ φ ) − m 2 ¯ L ( φ, ¯ φ, ∂ β φ, ∂ β ¯ µ ¯ φ, a α , ∂ β a α ) = ( D ∗ φ φ + L A . With an additional dynamical/kinetical term for the gauge fields, e.g. L A = − 1 4 F µν F 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 q , t ) = L ( Q , ˙ Q , t ) + ˙ L ( q , ˙ f ( q , Q , t ) , which induces the definition of a canonical transformation q − H ( q , p , t ) = P ˙ Q − H ( Q , P , t ) + ˙ p ˙ 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 f 1 ( q , Q , t ): p = ∂ f 1 ∂ q , P = − ∂ f 1 ∂ Q , H ( Q , P , t ) = H ( q , p , t ) + ∂ f 1 ∂ t . The corresponding symmetry relation is ∂ 2 f 1 ∂ p ∂ q ∂ Q = − ∂ P ∂ Q = ∂ 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 f 1 ( q , Q , t ), f 2 ( q , P , t ), f 3 ( p , Q , t ) and f 4 ( p , P , t ), which are connected by Legendre transformation, e.g. f 2 ( q , P , t ) = P Q + f 1 ( 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 f 1 ( q , Q , t ) f 2 ( q , P , t ) q p = ∂ f 1 ∂ q , P = − ∂ f 1 p = ∂ f 2 ∂ q , Q = ∂ f 2 ∂ Q ∂ P f 3 ( p , Q , t ) f 4 ( p , P , t ) p q = − ∂ f 3 ∂ p , P = − ∂ f 3 q = − ∂ f 4 ∂ p , Q = ∂ f 4 ∂ Q ∂ 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 2 mp 2 + m ω 2 1 q 2 . 0 H = 2 We choose f 1 ( q , Q ) = m ω 0 q 2 cot( Q ) , 2 which yields p = ∂ f 1 ∂ q = m ω 0 q cot( Q ) , 1 P = − ∂ f 1 ∂ Q = m ω 0 q 2 sin( Q ) . 2 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 � 2 P q = sin( Q ) , m ω 0 � p = 2 P m ω 0 cos( Q ) and the transformed Hamilton function reads H ( Q , P , t ) = ω 0 P 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 ∇ φ, x ) ≡ π ˙ H ( φ, π, � φ − L ( φ, ∂ α φ, x ) , where ∂ L π ≡ ∂ ( ∂ 0 φ ) . 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 ∂ L π µ ≡ ∂ ( ∂ µ φ ) which induces a covariant Legendre transformation for the Hamilton density H ( φ, π α , x ) ≡ π µ ∂ µ φ − L ( φ, ∂ α φ, 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 = − ∂ µ π µ , ∂ H ∂π µ = ∂ µ φ , ∂ µ H| expl = − ∂ µ L| 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 � ( ∂ µ φ )( ∂ µ φ ) − m 2 φ 2 � . 2 Consequently the canonically conjugate field is ∂ L π µ = ∂ ( ∂ µ φ ) = ( ∂ µ φ ) and we find the following expression for the Hamilton density H ( φ, π α , x ) = π µ ∂ µ φ − L ( φ, ∂ α φ, x ) = π µ ∂ µ φ − 1 � ( ∂ µ φ )( ∂ µ φ ) − m 2 φ 2 � 2 = 1 π µ π µ + m 2 φ 2 � � . 2 Adrian Königstein (ITP) Gauge theory in a Hamilton formalism 18.01.2016 21 / 45

Covariant Hamilton field theory Calculating the canonical equations − ∂ µ π µ = ∂ H ∂φ = m 2 φ , ∂ µ φ = ∂ H ∂π µ = π µ . Inserting one into the other we find the Klein-Gordon equation ( � + m 2 ) 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