So Called Vacuum Fluctuations as Correlation Functions. Robert D. Klauber August 23,

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1 So Called Vacuum Fluctuations as Correlation Functions Robert D. Klauber August, 6 Refs: Introduction to Quantum Effects in Gravit, Muhanov, V., and Winitzi, S. (Cambridge, 7 Student Friendl Quantum Field Theor, Klauber, R.D., (Sandtrove 5, nd ed, rd printing The Correlation Function in Quantum Field Theor. What It Is Mathematicall In Muhanov and Winitzi, pg. 5 unnumbered equation, and also, pg. 78, (6.48, the expectation value for vacuum fluctuations for a real scalar field is defined as a correlation function, in the vacuum, between two different spatial locations x and at the same time t. ( (,t (,t x x. ( Note that, if has local maxima at x and, will have a higher value there than if it has a local maximum at x and close to a null value at. In the first case, the value of at x is highl correlated with that at, whereas in the second case, it is not. So, in general, (x represents the correlation between different locations x and of the field. If represents a sinusoid, the highest correlations will occur between points separated b exact multiples of the wavelength of the sinusoid. A given pea is one wavelength from the next pea. But generall is not a pure sinusoid. In an case, of course, if x, we would get maximum (full correlation. Note further, that ( represents the vacuum expectation value (VEV of the bilinear (correlation operator (x,t (,t. It is usuall interpreted as the correlation one would expect to measure, on average, in the vacuum, between vacuum fields. We comment on this interpretation in Section 6. Using as ( ( ix ix ( x,t a e + a e d Muhanov and Winitzi (4., pg. 48 ( in (, we have ( / ω ix ix i i x ( a e + a e d ( a e a e d / / + ( ω.( ( ω. What It Means in the Phsical World The correlation function ( involves integration over all wave numbers and hence ma be visualized as correlating two points in phsical space for a wave composed of all (an infinite number wave length waves superimposed. Note that the amplitude of each wave is diminished b a factor of / ω. (For reasons wh this is true of Klein-Gordon waves, see Klauber, pg. 47, Normalization Factors and Relativistic Invariance of Probabilit sections. Non-relativistic [Schroedinger or macro/classical] tpe waves do not have this factor.

2 Deriving the Correlation Function. First Wa to Derive Correlation Function From (, we have (where destruction operators destro the vacuum, i.e., result in zero, ( ( ix i ix i ae a e + ae a e ω ω ix i ix i + ae a e + ae a e x d d ( ( ix i + ae a e ω ω ix i + + ae a e d d. ix i a e a e d d ω ω + ix i e e d d. ω ω since bra and et orthogonal ( (4 (5 ix i a a e e d d ω ω ( ( ( δ ( ix i a a e e d d. ω + ω Since a destros the vacuum, we have ( ( ( ix i x δ ( e e d d ω ω i( x i( ω ω t i ( x e d e e d ω ω (7 i ( x e d. ω ( ( We evaluate (7 using polar coordinates in space. (See Klauber, pg. 46, Fig We now tae the smbol, rather than the 4 momentum vector as it was used in exponents of ( to (7, and θ is the angle between and (x. Integration over is from to, θ from to, and from to. Thus, With (8, (7 becomes ( d sinθ ddθ x x cosθ. (8 ( ( ( i x x changes to i x cosθ x e d e d sin d d ( ω dependence ( ω θ θ x d cosθ i cosθ x ( i u e d cosθ d x e du d ( ω ( ω ( x ( tae as u i u i i x e x x e e d d ω i ( x ω i x ( (6 (9

3 or i i i i x x e e x e x e d d d ( ω i ( ω i x ω i x x i i( x e x e ( d ( d ( ω i x ω i ( x i i x e x e ( d d ( ω i x ω i x i i x e x e ( d + d ( ω i x ω i x i x e cos x d i sin x ( ω i ( ω i x d + x ( d ω i x since integrand is odd ( ( sin x ( d, ( ω x ( x all of which, from ( to (, Muhanov and Winitzi do in one step in the middle of page 5.. Second Wa to Derive Correlation Function From (5, we have ix i x a a e e d d ω. ( ω ( ( Aside: Note that a et in eigenstate, in space, is effectivel a Dirac delta function with a pea at. Aδ ( ( in space basis A is a normalization factor. (4 So, a number operator operating on (4 will leave the number of particles per unit in the et of -momentum, which is the delta function. That is, a ( a a ( N δ. (5 Note that we have a discrete state/et in (5 but we are woring with the continuous fields case math. Via analog with the discrete fields case, we can intuit that, given (5, then End of aside. Discrete Fields Case Continuous Fields Case ( ( ; + + δ a a n n n a ( ( a ; a n n n a δ Thus, with (6 in ( in we have. (6 See Klauber, pg. 59, (-8.

4 4 ix i x a a e e d d ω ω ( ( ( ( ix i a e e d d δ ω ω ( ( ix i e e d d δ ( δ ( ω ω ix i δ ( e e d d, ω ω which is the same as we found in the first row of (7. The rest follows as before in (7 to (.. Third Wa to Derive Correlation Function Starting again from (5, we have ix i x a a e e d d ω ω ( ( ix i a e e d d δ (, ω ω ( but now consider operating on the bra (not the et with a. This ields ( ( ( ix i ( x δ ( e e d d δ ( ω ω ix i δ ( δ ( e e d d ω ω ix i δ ( e e d d. ω ω which is the same as we found in the first row of (7. The rest follows as before in (7 to (. δ (7 (8 (9 (.4 Fourth Wa to Derive the Correlation Function Note that ( is simpl the particle form of the propagator as shown in Klauber, pg. 7, (-6 and elsewhere, where t t x t, i.e., ( ( ( ( for t t (particle x 6 and 8 in Klauber x where in the present case is real, i.e.,. ( In the reference, this is shown to equal i( x + e i ( x ( x ( d ( 9 in Klauber ( ω for t t x ( which for, equal times, as in ( where t t x t, is simpl e i( ωt ωt + i ( x e e d x ω ( (, (

5 5 as expressed in the last row of (7. Evaluating the Correlation Function We can consider the value of (, repeated below, for convenience.. Massless Field Case Since for the massless case, ( becomes sin x ( d, ( ω x ( x ω m where ± ± ω, (4 + sin x x d for m, i.e., ω ( x ( + sin x + cos d sin d x ( x x x x x ( cos. x (5 The cosine of infinit term can var between and + and depends on where (what value at ver large values we choose to cutoff the cosine oscillation and call it the end. (Its average value for a large number of random endpoints is zero. If we just ignore that term, we get an effective evaluation of the correlation between two points x and. ( x m L x (6 x L The correlation varies quadraticall with the inverse of L, the distance between x and. Note when x, the correlation is infinite, which maes some sense since a field at an point is completel correlated with itself at that point (at the same time.. Estimate for the Massive Field Case Relation ( is not so eas to evaluate when m, as then we have sin x d ( + m x, (7 sin d sin Ld L. ( x x ( L x + m + m ( x and there doesn t seem to be an integration table formula available for (7. Numerical integration appears in order for a precise solution at given L. However, we can get an estimate for (7, and Fig. can help demonstrate how. In the top row of Fig., where m, we can see that in the integration, all but the shaded area on the right side of the vertical axis will cancel out (ignoring the arbitrar cutoff we might chose on the far right and far left to approximate infinit. For the lower rows of Fig., the shaded region just to the right of the vertical axis approximates the total integral value, i.e., it is of the same order. So our job is to approximate the total integral b getting a good estimate of the region of integration of the curve from to.

6 6 L L m L ml m L L L m L m L L L m L m L Figure. Plots of the Integrand + m sinl vs L In the top row of the figure, the curve from to is half of a pure sine wave. In the lower rows, a half sine wave is a reasonable (of the same order approximation over that range. From the figure, we can see the amplitude of that approximate since wave is reduced for higher m or L. Note that the integration in (7 is over, but to simplif, the plots of Fig. are shown as over L. So given our approximations, we will estimate the integral of the entire integrand in (7 as that of a sine wave over from L of to, whose amplitude varies with m or L. To start, we will use the smbol f to denote the value of when L, and note that the integral over in the region of interest can be estimated with the help of the following. f cos L L L f L sin Ld ( cos + cos. (8 L L L For a sine curve of amplitude, as in the top row of Fig., (8 gives us the area under the curve from L to ( from zero to /L. For other amplitudes, as in the nd and rd rows of Fig., we need to multipl (8 b the maximum value (the sine curve amplitude of the integrand, which is the value of the amplitude at the mid point where L /. We label the value at this point as m. (The subscript m can stand for (local maximum, midpoint, or main contributing wave number to the area. So, Thus, the height of the function (integrand of (7 at m is the ml m. (9 L estimate of sine curve amplitude So using (8, (9, and ( to estimate (7, we have m ( + m m. (

7 or finall, + L sin L d L m sin L d ( L ( L + m + m ( 7 ( m L sin Ld L, ( L ( ( L L m + m + m L ( L L ml + m (. ( Note this checs with our earlier result (6 for m, as it should since our assumption of a pure sine wave over the region L to is precisel correct in that case.. Conclusions for Behavior.. With respect to m, L Note for m << L (large separation and/or small mass, effectivel varies inversel with the square of L. For m >> L, it effectivel varies with the inverse of the cube of L. As L, (L, and the correlation function, as defined in (, is infinite... With respect to wave number Note from Fig. that of all waves, the largest (main contributor to the correlation is the one with wave number m. And, we can relate to m, instead of L, via use of (9 in (, i.e., ( m L ( L L L m m m ml + +, ( or ( L 4 m m + m. (4.. With respect to m and m One commonl sees, as in Muhanov and Winitzi, pg. 5, (4.4, m taen as simpl in (4. Given that, one can surmise that for m << ( m reall, effectivel varies directl with the square of m ; for m >> ( m reall, directl with the cube of m. All of this Section, from pg. 5 to pg. 7, is done in one step in Muhanov and Winitzi in eq. ( Correlation Function for Discrete States Consider the discrete real field (which has finite volume boundar conditions

8 8 ( ( ( ix ix x a e + a ( e V ω in the correlation function ( (repeated below for convenience ( (,t (,t (5 x x. ( The effect of the creation/destruction operators is the same as for the continuous case of ( through (6 except the commutation relations give us a Kronecer delta instead of the Dirac delta function. ( ( + δ ix i x a a e e. (6 V ω ω As with the parallel continuous case, since a destros the vacuum, we have ( x ix i δ e e V ω ω ix i i ( ω ω t i e e e e ( x V ω V ω (7 i e ( x, V ω which parallels (7. In fact, as is well nown, in the limit V in the last row of (7, with appropriate boundar conditions, we have i ( x V i ( x x e ( e d V discrete to continuous x ω, (8 ω ( which corroborates our previous result (7. ( 5 More on the Phsical Interpretation of the Correlation Function Fig. can help us to get a better feeling for what we are reall measuring with our correlation function (L. Note from the LHS of Fig., we could find a correlation function for a single plane wave, if we desired. But that is not what we have done in prior sections, which is represented b the RHS of the figure. L x Finding correlation for one wave in eigenstate at distance L rd dimension suppressed peas of real part of plane waves of given wave number Finding correlation for all plane waves of all eigenstates at once of an infinite number of plane waves shown Figure. Showing What We Are Reall Finding with (L (The RHS Above x

9 9 We have started with the of (, which is a superposition of an infinite number of plane waves, in all directions, with all possible wave numbers. That is what we found in ( and other expressions for (L. We are determining the correlation between an two points x and, given the form of in (, an integration over all possible waves of eigenvector. Each eigenwave is modulated onl b the factor / waves contribute equall. Note that (L is the same for an two points a distance L apart. 6 Interpretation of Correlation Function for Vacuum Note several things.. The correlation function ( ω and otherwise all such x is independent of time. i.e., it is not fluctuating. It is the same for an time t. It ma be hard to see how it can represent fluctuations occurring in time in the vacuum or anwhere, as vacuum fluctuations are often portraed. But the underling field from which we determine is oscillating (fluctuating. It ma be argued to behave lie a tpical plane wave function for a free particle in non-relativistic quantum mechanics (NRQM of form which varies in time, but the probabilit densit iω t + i ψ state A e x e, (9 ψ stateψ state A A constant in time. (4 That is, the particle/state itself oscillates in time, but anthing we might detect lie particle position probabilit does not. In fact, phase (such as would change with time is irrelevant with respect to an measurements, so there seems to be little chance of detecting an sort of fluctuation in time of states existing in the vacuum. If the cannot be detected, one could argue that, for all practical purposes, the are not there.. The correlation function represents the correlation between locations in a particular wave/state/particle (in the vacuum or otherwise. It is not related to a pair of particles, which might be popping into or out of the vacuum, as is commonl depicted. Not onl is the correlation function representative of a single particle (comprised of all possible eigenstates superimposed, rather than a pair, it is static. It does not pop in and out of existence over time.. What seems to be happening in determining the correlation function appears different in each of the different approaches of Sections. to.4. The interpretation in each case is different. First Wa to Derive Correlation Function In Section., our et and bra remain vacuum states, and a non-zero correlation function arises in (6 from the operator commutation relation. That gives us a delta function, which is a number and has no effect on the vacuum et or bra. The delta function gives us the result of (7. So one could argue we are determining the correlation function for the vacuum, and if this is non-zero, it implies there are fields/particles in the vacuum. A pure zero number of fields in the vacuum should give rise to a zero correlation function. Indeed, the ½ quanta of the vacuum arise from the non-commutation relations as well. (See Klauber, pg. 54, (-54. So the source of the infinite (or at least enormous energ in the vacuum and the source of the correlation function are the same, the non-commutation of creation/destruction operators. The means for elimination of the ½ quanta energ in the vaccum suggested b Klauber (footnote pg. 5 also eliminates vacuum fluctuations. That is, it ields a zero value for (. It simpl incorporates mathematicall valid solutions to the field equations (Klein-Gordon here that have not been previousl incorporated into QFT.

10 Second Wa to Derive Correlation Function In Section., we do not use the commutation relations to convert the a a to a a plus a delta function. Instead we simpl act directl on the vacuum with a a. See (7 and (8. This implies we are creating a particle then destroing it all at the same time t. We get the same result. From this perspective, we are not measuring the correlation of fields in the vacuum, but the correlation between the state we create and the state we destro, which are the same. That is, the correlation function represents a correlation of a non-vacuum field with itself, and is not a measure of what is going on in the vacuum. Third Wa to Derive Correlation Function In Section., we again do not use the commutation relations, but emplo and a create a bra state via a. See (9 and (. a a to create a et state via From this perspective, we are again not measuring the correlation of fields in the vacuum, but the correlation between a created state and itself. That is, it is not a measure of what is going on in the vacuum. Fourth Wa to Derive Correlation Function In Section.4, the same correlation function is seen to be simpl the equal times Fenman propagator for a real field (both events are at time t. See ( to (. Propagators arise naturall in QFT via the Dson-Wics expansion, and represent virtual particles mediating interactions between real particles. Such propagators do not simpl pop in and out of the vacuum all alone, but are alwas lined to other real particles in interactions 4. Thus, one could argue that the correlation function, as defined in (, does not represent a characteristic of the vacuum existing in the absence of real world particles. As an aside, the correlation function ( with non equal times is not static but fluctuates in time. It is, of course, simpl the Fenman propagator. This ma be the reason propagators are sometimes referred to as vacuum fluctuations, or correlation functions. Conclusion: There are other interpretations to than that of a correlation function arising from vacuum fluctuations. a 4. Consider, instead of the real field of (, which is chargeless (i.e., the particle is its own antiparticle, a complex field (the particle has a distinct antiparticle,which has charge. Should the correlation function be or? One would expect the latter, as it is correlating the field with itself, rather than its complex conjugate. But that ends up with (a + b (a + b tpe terms in the correlation function (, rather than (a + a ( a + a tpe. That is, from (, ( a b ( a b ( a a a b b a b b terms of form (4 In that case all terms have a a, a b, b a or b b that either destro the vacuum (leave zero or form a et different from the bra, thereb ending up with zero. That is, there is zero vacuum correlation for a complex field (associated with particles lie electrons and quars. The term real field means the field is a real, not complex, function of space and time. The term real particle means a particle that is not virtual. 4 We are ignoring vacuum bubbles that do arise naturall from the Dson-Wics expansion. See Klauber, Section 8.4.8, pg. 4. Such vacuum bubbles alwas consist of three virtual particles, however, not a single one, as relates to.

11 Additionall, nothing lie (for complex fields appears anwhere in QFT. Not in the free theor; and not in the interacting theor. It plas no role in determining the transition amplitude for real world interactions. The transition amplitude, derived from the state equation of motion, has no terms in it. Conclusion: There is no correlation for a complex (charged field with itself, as determined b (, i.e.,. Further, plas no role in an transition amplitude. On the other hand, if we were to use instead of in (, then we are not correlating a field with itself, but with its complex conjugate. Is that a meaningful correlation? Still further, if we were to evaluate not a tpe. That is,, we would find a correlation onl between b tpe particles and ( a b ( a b ( a a a b b a bb terms of form (4 All of these ield zero except the (Similarl, b b term. So, no matter which of the was of Section we use to evaluate, the result will onl appl to b tpe particles and not a tpe, i.e., onl to antiparticles and not particles. would onl appl to particles and not antiparticles. Conclusion: It is hard to conclude we get a meaningful correlation function from not particles, are correlated. if onl antiparticles, and 5. Wouldn t we want the correlation to be with respect to probabilit densit at various points x and? Isn t the important thing how correlated what we measure (detect the particle is, rather than something we can t measure (the field itself? If we are taling about what we measure, should we be correlating probabilit densit ρ (x, t with ρ (, t, rather than (x, t with (, t? For a relativistic quantum scalar field theor, this is (4. (See Klauber, Sect..7 to Sect..8., pgs 6-64, and Sect. 5.6, (5-6, pg. 49. ( ρ i ɺ ɺ (4 For a real field, ρ is zero everwhere and for all time, and thus it is surel so for the vacuum. This led to the interpretation of (4 as charge densit. For a complex field, we get, for a single particle state (Klauber, pg. 6, (-9 ρ ρ V V ( Na ( Nb (, (44 where there are no ½ terms in the probabilit (charge densit ρ, so there is no contribution from the vacuum. Thus, ρ ρ ρ ( x,t (,t. (45 Conclusion: For the vacuum, there is no correlation between probabilit densities, i.e., from one point to another in what we should be able to measure experimentall. 6. (x,t (,t is not a measurable, in the usual quantum sense. (x,t (,t implies we can measure some quantit that is evaluated at two different points. But all measurables in QFT, lie energ densit, -momentum densit, charge densit, etc are measurable at a single point, i.e., related to (x,t (x,t at the same x at the same time t. Both field factors in an QFT bilinear operator similar in form to are evaluated at same event for measurable quantities. Conclusion: (x,t (,t is not a quantum measurable/observable.

12 7. It is interesting that, for the massless case of (6, the correlation falls off lie the photon radiation intensit from a charged point source radiating in all directions isotropicall. I am not sure what this might impl. 8. As an aside, one would thin the vacuum could not have an states in it, or a would not. 7 Possible Conclusions From Section 6, reasonable arguments could be made that the correlation function for a real scalar field, x x,t,t, does not represent fluctuations in fields residing in the vacuum, since there are ( ( ( different was to derive the final result, which comprise creation of states. For complex fields, i.e., fields associated with charged particles, the different possible definitions of lead x,t,t, inclusion of antiparticles, but not particles (for to zero (for ( ( ( (,t (,t ( (,t (,t x x, or inclusion of particles, but not antiparticles (for x x. Correlation of probabilit densities at two locations x and, at the same time t in the vacuum, ields zero. This is true for an form of, not just that of (. That is, it would be true if, for example, represented a Gaussian wave pacet. There is no vacuum correlation for probabilit densit. In fact, probabilit densit for the vacuum is identicall zero. Further arguments could also be made that does not represent a measurable quantit.