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Quantum Field Theory in Curved Spacetime: A brief

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<strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong><br />

In <strong>Curved</strong> <strong>Spacetime</strong><br />

José Roberto Vidal<br />

Universidad Autónoma de Madrid


Why QFT <strong>in</strong> curved ST?


Why QFT <strong>in</strong> curved ST?


Why QFT <strong>in</strong> curved ST?


Why QFT <strong>in</strong> curved ST?


Particles vs. fields<br />

A la We<strong>in</strong>berg


Particles vs. fields<br />

A la We<strong>in</strong>berg<br />

● Based on symmetries<br />

The Po<strong>in</strong>caré group<br />

tells the whole story:


Particles vs. fields<br />

A la We<strong>in</strong>berg<br />

● Based on symmetries<br />

● Multiparticle states<br />

We need the Fock space.<br />

And so on...


A la We<strong>in</strong>berg<br />

● Based on symmetries<br />

● Multiparticle states<br />

● <strong>Quantum</strong> fields are just<br />

a tool.<br />

Particles vs. fields<br />

Products of fields


Particles vs. fields<br />

Classical people:<br />

Poisson, Lagrange, Hamilton...<br />

A la Pesk<strong>in</strong>­Schroeder<br />

● We start with a classical<br />

system.


Particles vs. fields<br />

Diagonalization through<br />

Fourier transform:<br />

A la Pesk<strong>in</strong>­Schroeder<br />

● We start with a classical<br />

system.<br />

● Canonical quantization.


Particles vs. fields<br />

A la Pesk<strong>in</strong>­Schroeder<br />

● We start with a classical<br />

system.<br />

● Canonical quantization.<br />

● A beatiful metaphore.<br />

Particles are quantized<br />

excitations of the field


Part I<br />

QFT <strong>in</strong> curved ST is<br />

like Schw<strong>in</strong>ger effect!


1+1 FRW universe


1+1 FRW universe


In modes


No particles <strong>in</strong> the<br />

d<strong>in</strong>osaurs era<br />

In modes<br />

Complete and<br />

normalized basis<br />

And so on...


Out modes


No particles <strong>in</strong> the<br />

spacecrafts era<br />

Out modes<br />

And so on...<br />

Complete<br />

too!!


Particles from nowhere<br />

?<br />

?


Particles from nowhere


Particles from nowhere


Particles from nowhere


Particles from nowhere


Particles from nowhere


● Space plus time<br />

General formalism<br />

Clear separation<br />

between space an time<br />

??<br />

Suitable spacetimes


● Space plus time<br />

General formalism<br />

Clear separation<br />

between space an time


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

F<strong>in</strong>d a complete and<br />

normalized collection of<br />

modes.


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

F<strong>in</strong>d a complete and<br />

normalized collection of<br />

modes.


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

F<strong>in</strong>d a complete and<br />

normalized collection of<br />

modes.


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

● Quantize!<br />

We get automatically a<br />

well def<strong>in</strong>ed field operator<br />

plus a Hilbert space<br />

The<br />

pyramid<br />

once<br />

aga<strong>in</strong>


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

● Quantize!<br />

● Ambiguity<br />

Differents sets of modes<br />

give rise to differents<br />

“particles”.


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

● Quantize!<br />

● Ambiguity<br />

Differents sets of modes<br />

give rise to differents<br />

“particles”.<br />

Bogoliubov transformations


● Space plus time<br />

General formalism<br />

● Solve field equation<br />

● Quantize!<br />

● Ambiguity<br />

● And much more!<br />

Many other observables<br />

● Expectation values<br />

EM Tensor<br />

● Transition rates


Part II<br />

QFT <strong>in</strong> curved ST is<br />

not Schw<strong>in</strong>ger effect!


This is not Schw<strong>in</strong>ger effect!


What particles?!?


What particles?!?


Bonus<br />

Horizons and temperature


Bonus 1: M<strong>in</strong>kowski ST<br />

Accelerated observer


Bonus 2: de Sitter ST<br />

Time<br />

Horizon


Bonus 3: Black Holes


Unruh effect<br />

Accelerated observer


Unruh effect<br />

R<strong>in</strong>dler ST


Unruh effect<br />

And of course<br />

we already had


Unruh effect<br />

F<strong>in</strong>al surprise!


Hawk<strong>in</strong>g Radiation<br />

Two k<strong>in</strong>ds of observers/vacua<br />

Anyway<br />

or<br />

when compar<strong>in</strong>g or


● Birrel and Davies, <strong>Quantum</strong> <strong>Field</strong>s <strong>in</strong> <strong>Curved</strong> Space<br />

● V. Mukhanov, Introduction to <strong>Quantum</strong> Effects <strong>in</strong><br />

Gravity<br />

Further read<strong>in</strong>g<br />

● R. Wald, <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>in</strong> <strong>Curved</strong> <strong>Spacetime</strong><br />

and Black Hole Thermodynamics<br />

● S.A. Full<strong>in</strong>g, Aspects of <strong>Quantum</strong> <strong>Field</strong> <strong>Theory</strong> <strong>in</strong><br />

<strong>Curved</strong> Space­Time

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