* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Slides - WFU Physics
Quantum group wikipedia , lookup
Quantum teleportation wikipedia , lookup
Renormalization group wikipedia , lookup
Quantum entanglement wikipedia , lookup
Scalar field theory wikipedia , lookup
Bell's theorem wikipedia , lookup
Matter wave wikipedia , lookup
Quantum key distribution wikipedia , lookup
Path integral formulation wikipedia , lookup
Elementary particle wikipedia , lookup
Dirac equation wikipedia , lookup
Renormalization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Particle in a box wikipedia , lookup
Density matrix wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum state wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
EPR paradox wikipedia , lookup
Canonical quantization wikipedia , lookup
History of quantum field theory wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Strangeness production wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Hidden variable theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
? Eric Carlson Einstein’s Equation Relates the shape of spacetime to the stuff that’s in it Curvature G 8 GT of spacetime jx T jy jz jx S xx jy S xy S yx S zx S yy S zy jz S xz S yz S zz Energy density (and other quantities in T) need to include quantum effects i T x x x x x 2 The problem: We don’t have a quantum theory of gravity G 8 GT Semi-Classical Gravity G 8 GTT •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 Why it might make sense •There are lots of particles we know how to do quantum mechanics on Particle Higgs symbols spin H 0 Electron e Electron neutrino e Up quark uuu Down quark ddd Muon Muon neutrino Up quark ccc Down quark sss Tau Tau neutrino Top quark ttt Bottom quark bbb d.o.f. 1 ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ 4 2 12 12 4 2 12 12 4 2 12 12 2 16 6 3 2 Photon Gluon W-boson Z-boson gggggggg W Z 1 1 1 1 Graviton h 2 118 2 Spin ½ fields near wormholes x r x=0 r=r0 “throat” What does the asymptotic energy density look like? •Wormholes connect distant points in space •Wormholes require negative energy density •It is possible (likely) that wormholes would have negative energy density •Naive use of the “analytic approximation” predicted that the energy density would fall as 1/r6 at large r •Other arguments predicted 1/r5 The Method 1. Convert classical equations for free fields to curved spacetime 0 0 2. Solve Green’s function equations in curved spacetime S x, x 4 x x ' 3. Use Green’s functions to calculate expectation value of T ) S E S Ec T x unren 14 lim Im Tr ( x x ' g ' ' S E S Ec I x, x ' 4. Renormalize to get rid of infinities T x ren T x modes T x WKBfin T x analytic Computational approach 1. Solve lots of coupled differential equations G , j x j 12 F , j G , j r x f x F x do i=1,imax h=htot/nseq(i) zold=z znew=z+h*dzdx xx=x+h 1 twoh=h+h 2 do j=2,nseq(i) , j , j , j 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))) 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 f x G j F r x 2. Add together all the modes 3. Integrate over frequency 4. Add other terms What you get R. Chainani, 9/21/09 0.03 0.025 0.02 Tt t 0.015 T 2r5Tµ/b 0.01 Trr 0.005 0 1 2 3 4 5 6 7 8 -0.005 -0.01 -0.015 -0.02 2 dr 2 2 2 2 2 ds 2 dt 2 r d r sin d 1 b2 r 2 9 10 r /b 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