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Lecture 1 of 4: Zero-Point Energy

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Lecture 1 of 4: Zero-Point Energy Victoria University of Wellington Te Whare W¯anangao te Upoko¯ o te Ika a Maui VUW Lecture 1 of 4: Zero-point energy Matt Visser AotearoaFP16: Quantum Field Theory Victoria University of Wellington 12{16 December 2016 Matt Visser (VUW) L1 of 4: Zero-point energy 1 / 41 Outline: VUW Any quantum simple harmonic oscillator has zero-point energy: 1 E = !: 0 2~ There are many physical situations in which this zero-point energy is undoubtedly real and physical. There are other physical situations in which this zero-point energy has a much more ambiguous status. Lead in to my subsequent lectures: Zero-point energies in QFT. Casimir energies. Sakharov-style induced gravity. Key message: | Whenever possible concentrate on physical energy differences. Matt Visser (VUW) L1 of 4: Zero-point energy 2 / 41 Outline: VUW 1 Introduction 2 Bound states 3 Lamb shift 4 Casimir effect 5 Spontaneous emission 6 Zero-point energy density 7 Significance of the Planck scale 8 Worst prediction in particle physics? 9 Conclusions Matt Visser (VUW) L1 of 4: Zero-point energy 3 / 41 Introduction: VUW Introduction Matt Visser (VUW) L1 of 4: Zero-point energy 4 / 41 Introduction: VUW Theorem For the quantum SHO: 1 E = n + ! n 2 ~ Theorem For the quantum SHO the lowest energy state is: 1 E = ! 0 2 ~ This is often called the zero-point energy. What is the zero-point energy good for? Matt Visser (VUW) L1 of 4: Zero-point energy 5 / 41 Bound states VUW Bound states Matt Visser (VUW) L1 of 4: Zero-point energy 6 / 41 Bound states VUW Consider a non-positive potential that asymptotes to zero: V (x) ≤ 0; V (±∞) ! 0: Approximate the potential around its local minima by: 1 V (x) = −∆ + V 00 (x − x )2 + ::: 2 0 0 Approximate bound state energies are: r 1 V 00 E ≈ −∆ + n + 0 ; E < 0: n 2 ~ m n Matt Visser (VUW) L1 of 4: Zero-point energy 7 / 41 Bound states VUW The number of bound states is approximately: ∆ 1 ∆ r m 1 N ≈ + = 00 + : ~! 2 ~ V0 2 To get at least one bound state (n = 0, N = 1) we need: r 1 1 V 00 ∆ ! = 0 : & 2~ 2~ m To prevent bound states: r 1 1 V 00 ∆ ! = 0 : . 2~ 2~ m Zero-point energy controls the number of bound states... (at least approximately)... Matt Visser (VUW) L1 of 4: Zero-point energy 8 / 41 Bound states VUW Example| sech 2 potential (simplified P¨oschl{Teller potential): ∆ ∆ V (x) = − ≈ −∆ + x2 + ::: cosh2(x=a) a2 Estimate number of bound states using ZPE: p ! ∆ 1 ∆ r m 1 1 2∆ma2 N ≈ + = 00 + = + 1 : ~! 2 ~ V0 2 2 ~ Matt Visser (VUW) L1 of 4: Zero-point energy 9 / 41 Bound states VUW Define: 2 ∆ = ~ λ(λ − 1) 2ma2 Exact eigen-energies (standard exercise): 2 E = − ~ (λ − 1 − n)2; 0 ≤ n ≤ λ − 1 n 2ma2 Exact number of bound states: " p !# 1 2∆ma2 N = floor(λ) = floor + 1 2 ~ Number of bound states, (if not precise eigen-energies), agrees with the estimate from zero-point energy... Matt Visser (VUW) L1 of 4: Zero-point energy 10 / 41 Bound states VUW There are many excruciatingly precise and technical theorems on the existence/non-existence/number of bound states... But the zero-point energy argument gets to the heart of the matter quickly and incisively... Zero-point energies are physically real... Matt Visser (VUW) L1 of 4: Zero-point energy 11 / 41 Lamb shift: VUW Lamb shift Matt Visser (VUW) L1 of 4: Zero-point energy 12 / 41 Lamb shift VUW To see that there isa non-zero Lamb shift is deceptively easy... Energy difference: 2 2 ∆E = E S1=2 − E P1=2 : To estimate the numerical value is much more subtle... Key trick: Assume electrons are “buffeted” by zero-point fluctuations and write: e2 e2 e2 V = ! = exp (~rzpf · r) r j~r + ~rzpf j r 1 e2 ≈ 1 + (~r · r) + (~r · r)2 + ::: zpf 2 zpf r Matt Visser (VUW) L1 of 4: Zero-point energy 13 / 41 Aside: Taylor series: VUW Aside: Taylor series: exp(~a · r~ ) f (~x) = f (~x + ~a) You may not have seen it presented this way before... But this really is Taylor's theorem for C ! (analytic) functions... Simply expand the exponential on the LHS... Matt Visser (VUW) L1 of 4: Zero-point energy 14 / 41 Lamb shift: VUW Average over zero point fluctuations, making minimal (isotropy) assumptions: 1 h~r i = 0; h~r ⊗ ~r i = h(~r )2i I : zpf zpf zpf 3 zpf 3×3 Then: 1 e2 hV i ≈ 1 + h(~r )2ir2 + ::: 6 zpf r e2 1 ≈ − h(~r )2i 4π δ3(~r) + ::: r 6 zpf This implies s-wave slightly less tightly bound than p-wave... 2π ∆E = h(~r )2i j (0)j2::: 3 zpf Matt Visser (VUW) L1 of 4: Zero-point energy 15 / 41 Lamb shift: VUW 2 So \all" you now need to do is to estimate h(~rzpf ) i.... Have fun... With a little work: 1 2 h(~r )2i ≈ α ~ ln(4/α): zpf 2π2 mc With a little more work α5mc2 1 ∆E = E 2S − E 2P ≈ ln : 1=2 1=2 6π πα Clear message: Zero-point fluctuations have real physical effects... Matt Visser (VUW) L1 of 4: Zero-point energy 16 / 41 Casimir effect: VUW Casimir effect Matt Visser (VUW) L1 of 4: Zero-point energy 17 / 41 Casimir effect: VUW Introduce some conducting boundaries... Solve for the eigen-frequencies !n. Formally sum: X 1 E = ! Casimir 2~ n n Being more careful, compare two suitably chosen systems ... Calculate Casimir energy differences: X 1 ∆E = (! − !∗) Casimir 2~ n n n See Lecture 3 for theory details... Review the experimental situation... Zero-point energies are real... Matt Visser (VUW) L1 of 4: Zero-point energy 18 / 41 Spontaneous emission: VUW Spontaneous emission Matt Visser (VUW) L1 of 4: Zero-point energy 19 / 41 Spontaneous emission VUW There is an argument, based on the Einstein A and B coefficients, that the zero-point energies simply have to be physically real in order to be consistent with the experimentally observed phenomenon of spontaneous emission.... (Wikipedia says so, it must be true...) (Details mercifully suppressed for now...) Matt Visser (VUW) L1 of 4: Zero-point energy 20 / 41 Spontaneous emission VUW Conclusion: For each 3-vector ~k, and each polarization mode, the electromagnetic field is associated with a SHO of frequency ! = c j~kj... So for n photons in this SHO one has 1 E = n + ! n 2 ~ The total energy (per polarization) in the electromagnetic field is then V Z 1 E = d3~k n(~k) + !(k~) tot (2π)3 2 ~ Even in the quantum vacuum state one has V Z 1 E = d3~k !(~k) tot (2π)3 2~ Matt Visser (VUW) L1 of 4: Zero-point energy 21 / 41 Spontaneous emission VUW But the integral Z 1 Z 1 d3~k !(~k) = d3~k cjk~j 2~ 2~ diverges... WTF? We need to think about this... Carefully... Matt Visser (VUW) L1 of 4: Zero-point energy 22 / 41 Zero-point energy density: VUW Zero-point energy density Matt Visser (VUW) L1 of 4: Zero-point energy 23 / 41 Zero-point energy density: VUW The zero-point energy density in the electromagnetic field is: c Z d3k ρ = 2 × ~ jk~j: zpe 2 (2π)3 Introduce a cutoff: Z K 3 Z K ~c d k ~ ~c 3 ~c 4 ρzpe = 2 × 3 jkj = 2 dk k = 2 K 2 0 (2π) 2π 0 8π Extremely naively, (much better will be done tomorrow in Lecture 2), let us assert that the cutoff is at the Planck scale... Matt Visser (VUW) L1 of 4: Zero-point energy 24 / 41 Significance of the Planck scale: VUW Significance of the Planck scale Matt Visser (VUW) L1 of 4: Zero-point energy 25 / 41 Significance of the Planck scale: VUW Quantum mechanics tells us that an elementary particle of mass m can be reasonably easily localized within a distance λ = ~ Compton mc known as the Compton wavelength. Matt Visser (VUW) L1 of 4: Zero-point energy 26 / 41 Significance of the Planck scale: VUW 6Compton wavelength - Mass Matt Visser (VUW) L1 of 4: Zero-point energy 27 / 41 Significance of the Planck scale: VUW Classical gravity tells us that a particle of mass m will disappear down a black hole if the particle is smaller than its Schwarzschild radius 2Gm r = : Schwarzschild c2 Matt Visser (VUW) L1 of 4: Zero-point energy 28 / 41 Significance of the Planck scale: VUW 6Schwarzschild radius - Mass Matt Visser (VUW) L1 of 4: Zero-point energy 29 / 41 Significance of the Planck scale: VUW A heavy enough elementary particle should disappear down its own little black hole. We expect this to happen when the Compton wavelength equals the Schwarzschild radius. Set λCompton = rSchwarzschild... Solve for the mass m of the particle... This defines the Planck mass: r c M = ~ : Planck G If we plot the Compton wavelength as a function of mass, and the Schwarzschild radius as a function of mass, the Planck mass is the place that the two graphs cross. Matt Visser (VUW) L1 of 4: Zero-point energy 30 / 41 Significance of the Planck scale: VUW 6Distance `Planck - mPlanck Mass Matt Visser (VUW) L1 of 4: Zero-point energy 31 / 41 Significance of the Planck scale: VUW Once we have the Planck mass, the Planck energy is easy: 2 Take EPlanck = mPlanckc to get r c5 E = ~ : Planck G The Compton wavelength of a Planck mass particle, λPlanck = ~=(mPlanckc), is defined to be the Planck length p `Planck = ~cG: Finally the Planck time is defined to be the time required for light to travel one Planck length, TPlanck = `Planck=c, so that r G T = ~ : Planck c Matt Visser (VUW) L1 of 4: Zero-point energy 32 / 41 Significance of the Planck scale: VUW From the theorists' perspective, one of the most frustrating aspects of our times is that all the interesting physics, (interesting from the point of view of quantum gravity that is), seems to be taking place at or above the Planck scale | but our current technology is simply not up to the task of building a Planck scale accelerator.
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