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Gauge Symmetry

Gauge Symmetry. Mr Finn Honors Physics March 2012. Electric Potential, V. Electric Potential V = QPE/q  QPE = QPE f - QPE i = QPE( ) - QPE(d) Define QPE()  0 QPE = kQq/d Choice of where QPE  0 is arbitrary This choice has NO physical implications

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Gauge Symmetry

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  1. Gauge Symmetry Mr Finn Honors Physics March 2012

  2. Electric Potential, V • Electric Potential V = QPE/q • QPE = • QPEf - QPEi = QPE() - QPE(d) • Define QPE()  0 • QPE = kQq/d • Choice of where QPE  0 is arbitrary • This choice has NO physical implications • Electric field = -V/x depends on only differences, not values for V Recall V can be used to derive all relevant quantities.

  3. Symmetry • Observables remain constant after some “change” occurs • Types of symmetries • discrete • continuous • “Gauge” symmetry  V replaced by V+C • change in “scale” or C = continuous • global vs. local symmetry • misnomer! V = function is like a triangle Adding C is like rotating/reflecting a triangle Result is physically the same

  4. A C B C A B Triangles • Rotate equilateral triangle by 120° • vertices identical, labels irrelevant • physically original and rotated triangles are identical • Potential and new gauge • gauge irrelevant • fields, forces, changes in energy … all the same • physically V, V+C are identical

  5. Indirect Proof • Two fundamental principles • gauge symmetry of electric forces (GS): V = V + C • conservation of electric charge (CoQ) • GS  CoQ • Theory is that GS implies CoQ • Proof • Assume GS to be true • Assume CoQ to be false • Derive contradiction or that GS is false • GS cannot be both true and false • Premise that CoQ is false must be itself false and CoQ must be true • Recall: • GS = follows from Coulomb’s law • CoQ = empirically verified by Franklin • Problem: Are these “truths” related?

  6. destroy +1 nC of charge + + move charge create +1 nC of charge Assume we can Violate Conservation of Charge Create charge, move it, destroy it +60 V Equipotential lines +20 V Without also Violating Gauge Symmetry

  7. Energy • Create +1 nC • E1 = qV1 = (1nC)(20 J/C) = 20  10-9 J • Move charge • W = q V = (1 nC)(60-20) J/C = 40  10-9 J • Destroy +1 nC • E2 = -qV2 = -(1nC)(60 J/C) = - 60  10-9 J • negative = released, opposite of absorbed • Energy is conserved Ignore energy to create mass

  8. +160 V + + +120 V Contradiction • Add 100 V to all equipotential lines • Change “gauge” and nothing observable should be different • Energy still conserved • But energy to create charge changes! • This is observable • Violate GS • Assumed symmetry is broken

  9. Charge must be conserved • Otherwise, “gauge” (constant added to electric potential) can be measured • But gauge is irrelevant - only differences in electric potential are observable • Gauge symmetry  conservation of charge • Assuming energy is conserved Or: Gauge symmetry and charge conservation are logically connected  one implies the other.

  10. Conclusion • Mathematical structure of electrical interactions • their ability to be represented by scalar potential field V • which obeys a global gauge symmetry • Logically implies that the property that creates the interaction must itself be a conserved quantity • electric charge can be neither created nor destroyed

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