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Eric Carlson

Quantum Fields. in. Curved Spacetime. ?. Eric Carlson. Curvature of spacetime. Einstein’s Equation. Relates the shape of spacetime to the stuff that’s in it. Quantum. Classical. Quantum Fields. Energy density (and other quantities in T  ) need to include quantum effects.

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Eric Carlson

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  1. Quantum Fields in Curved Spacetime ? Eric Carlson

  2. Curvature of spacetime Einstein’s Equation Relates the shape of spacetimeto the stuff that’s in it

  3. Quantum Classical Quantum Fields Energy density (and other quantitiesin T) need to include quantum effects The problem: We don’t havea quantum theory of gravity

  4. Semi-Classical Gravity • Calculate T including quantum effects in curved spacetime • Replace T by its expectation value • Find the shape of spacetime from Einstein’s equation (semi-classical version) • Repeat until it converges

  5. Why it might make sense 118 2 Particlesymbolsspind.o.f.Higgs H 0 1 Electron e ½ 4Electron neutrino e ½ 2Up quark uuu ½ 12Down quark ddd ½ 12Muon  ½ 4Muon neutrino  ½ 2Up quark ccc ½ 12Down quark sss ½ 12Tau  ½ 4Tau neutrino  ½ 2Top quark ttt ½ 12Bottom quark bbb ½ 12 Photon 1 2Gluon gggggggg1 16W-boson W 1 6Z-boson Z 1 3 Graviton h 2 2 • There are lots of particles we know how to do quantum mechanics on

  6. Spin ½ fields near wormholes • Wormholes connect distant points in space • Wormholes require negative energy density • It is possible (likely) that wormholes would have negative energy density x r r=r0 “throat” x=0 • Naive use of the “analytic approximation” predicted that the energy density would fall as 1/r6 at large r • Other arguments predicted 1/r5 What does the asymptotic energy density look like?

  7. The Method 1. Convert classical equations for free fields to curved spacetime 2. Solve Green’s function equations in curved spacetime 3. Use Green’s functions to calculate expectation value of T 4. Renormalize to get rid of infinities

  8. Computational approach 1. Solve lots of coupled differential equations do i=1,imax h=htot/nseq(i) zold=z znew=z+h*dzdx xx=x+h twoh=h+h do j=2,nseq(i) call rf(xx,r,f) swap=zold+twoh*(ell*(1.q0-znew**2)/r + -2*omega*znew/sqrt(f)) zold=znew znew=swap xx=xx+h enddo call rf(xx,r,f) zold=half*(zold+znew+h*(ell*(1.q0-znew**2)/r + -2*omega*znew/sqrt(f))) 2. Add together all the modes 3. Integrate over frequency 4. Add other terms do i=1,ihi a1=l*omega/r(i)*(zp(i)+zq(i))/(zp(i)-zq(i)) a2=l*sqrt(f(i))*l/r(i)**2*(1-zp(i)*zq(i))/(zp(i)-zq(i)) w=sqrt(omega**2*r(i)**2+l**2*f(i)) a1w=t10(i)*l*omega**2/w a2w=t20(i)*qfloat(l)**3/w do k=1,lev do j=1,2*k a1w=a1w+l*t1(i,k,j)*l*omega**(2*j)*l/w**(2*j+2*k+1) a2w=a2w+l*t2(i,k,j)*l*omega**(2*j)*l/w**(2*j+2*k+1) enddo enddo

  9. What you get R. Chainani, 9/21/09 Ttt T Trr 2r5Tµ/b

  10. What you need to do this research • Undergraduates: • Strong Mathematical Background • Computer Skills Helpful • Maple or Mathematica Experience • Graduates: • Graduate Quantum Mechanics • General Relativity – must be arranged • Quantum Field Theory - must be arranged

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