Quantum Field Theory in Curved Spacetime: A brief
Quantum Field Theory in Curved Spacetime: A brief
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