**In Flat Spacetime**

Consider a scalar field in flat spacetime (equation 37 in The Essential Quantum Field Theory)

(1) Φ

_{k}= ∫d

^{3}

**k**(2π)

^{-3}(2ω

_{k})

^{-½}[a

_{k}e

^{ik∙x}+ a

_{k}

^{†}e

^{-ik∙x}]

We define the quantum fluctuations as,

(2) δΦ

_{k}≡ (< |Φ

_{k}|

^{2}>)

^{½}

Substitute 1 into 2,

(3) δΦ

_{k}~ (ω

_{k})

^{-½}

Field values cannot be measured at a point - in a realistic experiment, only their values averaged over a finite region of space can be measured. Consider the average value of a field Φ(

**x**) in a cubed-shape region of volume L

^{3}.

(4) Φ

_{L}≡ L

^{-3}∫

_{-L/2}

^{+L/2}dx ∫

_{-L/2}

^{+L/2}dy ∫

_{-L/2}

^{+L/2}dz Φ(

**x**)

For k = L

^{-1}

(5) δΦ

_{L}~ {(δΦ

_{k})

^{2}k

^{3}}

^{½}(See Appendix A)

Using equation 1, we get,

(6) δΦ

_{L}~ ( k

_{L}

^{3}/ω

_{kL})

^{½}, k

_{L}≡ L

^{-1}

We see that δΦ

_{L}diverges for small L (large k, large mass), decays for large L (small k, small mass). Quantum vacuum fluctuations have been observed in the spontaneous emission of radiation by atoms, the Lamb shift, and the Casimir effect. These observations cannot be explained by any other known physics.

**In Curved Space-Time**

Recall equation 21 from Quantum Fields in Curved Space-Time , written below,

(7) Χ(

**x**,η) = 2

^{-½}∫d

^{3}

**k**(2π)

^{-3/2}[e

^{ik∙x}v*

_{k}(η)a

_{k}

^{-}

+ e

Given a quantum state, the amplitude of quantum fluctuations is always well-defined irrespective of whether the particle interpretation of the field is available (see discussion after equation 47 in Quantum Fields in Curved Space-Time )
^{-ik∙x}v_{k}(η)a_{k}^{+}]Let us consider the equal-time correlation function for the vacuum state

(8) < 0|Χ(x,η)Χ(y,η)|0 > ~ k

^{3}(sinkL)/(kL)|v

_{k}(η)|

^{2}(See appendix B)

We can see that at k ~ L

^{-1}

(9) (sinkL)/(kL) ~ 1

(10) Therefore, < 0|Χ(x,η)Χ(y,η)|0 > ~ k

^{3}|v

_{k}|

^{2}

**Fluctuations of Spatially Averaged Fields**

There is another way to characterize the average fluctuations in a box of size L. This is done by using a

*window-averaged operator*,

(11) X

_{L}(η) = ∫ X(

**x**,η)W

_{L}(

**x**) d

^{3}

**x**

This function must satisfy the condition,

(12) ∫ W

_{L}(

**x**) d

^{3}

**x**= 1

A typical example of a window function is the Gaussian function,

(13) W

_{L}(

**x**) = (2π)

^{-3/2}L

^{-3}exp{ -|

**x**|

^{2}/(2L

^{2})}

This has the following scaling property,

(14) W

_{L'}(

**x**) = (L

^{3}/L'

^{3})W

_{L}(

**x**)((L/L')

**x**)

The Fourier image is then defined as,

(15) w(

**k**L) = ∫W

_{L}(

**x**)e

^{-ik∙x}d

^{3}

**x**

Which satifies,

(16) w|

_{k=0}= 1 and decays rapidly for |

**k**|≥ L

^{-1}.

We can now calculate,

(17)δX

_{L}

^{2}(η)= < 0| [∫ X(

**x**,η)W

_{L}(

**x**) d

^{3}

**x**]

^{2}|0 >, (from equ. 11)

Using equation 7 and 15, this reduces to,

(18) δX

_{L}

^{2}(η) ~ ∫ d

^{3}

**k**|w(

**k**L)|

^{2}|v

_{k}|

^{2}

Applying conditions 16, we get

(19) δX

_{L}

^{2}(η) ~ k

^{3}|v

_{k}|

^{2}

Which is the same result for the two-point correlation we found in curved space-time (equation 10). Therefore we define,

(20) δ(k) ≡ (2π)

^{-1}k

^{3/2}|v

_{k}|

as the

*spectrum of quantum fluctuations*, which characterizes the typical amplitude of quantum fluctuations on scales L.

**Complimentary Notes**: In a de Sitter space, the spectrum of quantum fluctuations ~ as the Hubble constant. One can show that the quantization of gravitational waves in an expanding universe is reduced to the problem of quantizing a massless scalar field. Furthermore, the production of primordial inhomogeneities can be found in a nearly scale-invariant spectrum.

**Appendix A**

(A1) Φ

_{L}= L

^{-3}∫

_{L3 }Φ(

**x**)d

^{3}

**x**

~ L

The integral over d^{-3}∫_{L3}d^{3}**x**∫d^{3}**k**e^{ik∙x}Φ_{k}^{3}

**x**can be computed first. Consider that (

**x**= x,y,z), then the x-component of that integral is,

(A2) ∫

_{-L/2}

^{+L/2 }dx e

^{ikxx}= ∫

_{-1}

^{1}d(cosθ)e

^{ikxLcosθ}

= (2/k

_{x}L) sin(k_{x}L/2) ≡ f(k

We get similar expression for the y and z components. The expectation value for Φ_{x})_{L}

^{2}is

(A3) < Φ

_{L}

^{2}> ~ ∫d

^{3}

**k**d

^{3}

**k'**< Φ

_{k}Φ

_{k'}>f(k

_{x})f(k

_{y})f(k

_{z})f(k'

_{x})f(k'

_{y})f(k'

_{z})

(A4) But < Φ

_{k}Φ

_{k'}> = (δΦ

_{k})

^{2}δ(

**k**+

**k**') (equ.2)

(A5) Therefore, < Φ

_{L}

^{2}> ~ ∫d

^{3}

**k**(δΦ

_{k})

^{2}{f(k

_{x})f(k

_{y})f(k

_{z})}

^{2}

The function f(k) is of order 1 for |kL| ≤ 1, and very smal for |kL| >>1. We may take δΦ

_{k}to be constant over the integration. Then

(A6) < Φ

_{L}

^{2}> ~ ∫d

^{3}

**k**(δΦ

_{k})

^{2}~ k

^{3}(δΦ

_{k})

^{2}, k = L

^{-1}

**Appendix B**

From equation 7 reproduced below,

(B1) Χ(

**x**,η) = 2

^{-½}∫d

^{3}

**k**(2π)

^{-3/2}[e

^{ik∙x}v*

_{k}(η)a

_{k}

^{-}

+ e

(B2) Calculating
^{-ik∙x}v_{k}(η)a_{k}^{+}]< 0|Χ(x,η)Χ(y,η)|0 > = < 0| 2

^{-½}∫d

^{3}

**k**(2π)

^{-3/2}[e

^{ik∙x}v*

_{k}(η)a

_{k}

^{-}+ e

^{-ik∙x}v

_{k}(η)a

_{k}

^{+}]

*x*2

^{-½}∫d

^{3}

**k'**(2π)

^{-3/2}[e

^{ik'∙y}v*

_{k'}(η)a

_{k'}

^{-}+ e

^{-ik'∙y}v

_{k'}(η)a

_{k'}

^{+}]|0 >

= < 0|2

^{-1}(2π)^{-3}∫d^{3}**k**∫d^{3}**k'**[e^{ik∙x}v*_{k}(η)a_{k}^{-}e^{ik'∙y}v*_{k'}(η)a_{k'}^{-}+ e^{ik∙x}v*_{k}(η)a_{k}^{-}e^{-ik'∙y}v_{k'}(η)a_{k'}^{+} + e

Recall that,
^{-ik∙x}v_{k}(η)a_{k}^{+}e^{ik'∙y}v*_{k'}(η)a_{k'}^{-}+ e^{-ik∙x}v_{k}(η)a_{k}^{+}e^{-ik'∙y}v_{k'}(η)a_{k'}^{+}] |0 >(B3) a

_{k}

^{-}|0 > = < 0|a

_{k}

^{+}= 0

The only surviving term in B2 is,

(B4) < 0|Χ(x,η)Χ(y,η)|0 > = < 0|2

^{-1}(2π)

^{-3}∫d

^{3}

**k**∫d

^{3}

**k'**e

^{ i(k∙x - k'∙y)}v*

_{k}(η)a

_{k}

^{-}v

_{k'}(η)a

_{k'}

^{+}|0 >

Recall equation 25 from Quantum Fields in Curved Space-Time , written below,

(B5) [a

_{k}

^{-},a

_{k'}

^{+}] = δ(

**k**−

**k'**), [a

_{k}

^{-},a

_{k'}

^{-}]= 0, [a

_{k}

^{+},a

_{k'}

^{+}]= 0

From the first, we have

(B6) a

_{k}

^{-}a

_{k'}

^{+}= δ(

**k**−

**k'**) + a

_{k'}

^{+}a

_{k}

^{-}

Substituting B6 into B4, taking care of B3

(B7) < 0|Χ(x,η)Χ(y,η)|0 > = < 0|2

^{-1}(2π)

^{-3}∫d

^{3}

**k**∫d

^{3}

**k'**e

^{i(k∙x - k'∙y)}δ(

**k**−

**k'**)v*

_{k}(η)v

_{k'}(η) |0 >

= 2

^{-1}(2π)^{-3}∫d^{3}**k**e^{ik∙(x - y)}v*_{k}(η)v_{k}(η) = 2

^{-1}(2π)^{-3}∫d^{3}**k**e^{ik∙L}v*_{k}(η)v_{k}(η), where L = |**x**-**y**| ~ k

^{3}(sinkL)/(kL)|v_{k}(η)|^{2}, (Using equation A2)