Gaussian Measures and the QM Oscillator
Sabrina Gonzalez Pasterski
(Dated: April 20, 2014)
In this paper, I show how probability densities associated with a Gaussian field can be expressed
in terms of the Boltzmann heat kernel. The N  2 calculations are based o↵ of the work of Arthur
Ja↵e, while the proof of his postulate for general N is original.
In “Fields with a Gaussian Measure,” I found:
R QN
⇢t (x) ⌘
xi )dµc
i=1 ( (ti )
=p
where Cij = C(ti
the operator:
H0 =
1
e
(2⇡)n detC
tj ) =
1
2
✓
1 >
2x C
1
x
1
m|ti tj |
.
2m e
2
d
+ m2 x2
dx2
(1)
Now, consider
◆
m
(2)
⇣ m ⌘1/4
⇡
e
mx2
2
1
For N = 1, ⇢t (x) = ⌦0 (x)2 . When N > 1, it is convenient to define C = 2mC so that Cij = e m|ti tj | .
Then:
⇢t (x) =
(5)
Here, I will consider t1 < ... < tN . For N = 2 explicitly
inverting
✓
◆
1
e m(t2 t1 )
C2 ⌘
(6)
e m(t2 t1 ) 1
gives an expression for ⇢ in terms of B:
⇢t1 ,t2 (x1 , x2 ) = ⌦0 (x1 )Bt2
t1 (x1 , x2 )⌦0 (x2 )
N
(CN
1)
), leads to an ex(CN1 1 v)
µ
1
µ
1 v)
(7)
1
A
(9)
where µ = 1 v > (CN1 1 v). Rather than inverting CN 1 ,
my expression for ⇢N in terms of ⇢N 1 will only need
the product (CN1 1 v). Because the inverse exists, it is
equivalent to finding ⇠ such that v = CN 1 ⇠. Since the
last column of CN 1 is (e m(tN 1 t1 ) ...1), I find that:
(3)
For t > 0, the Boltzmann integral kernel gives the evolution:
Z 1
e tH0 f (x) =
Bt (x, x0 )f (x0 )dx0
(4)
>C 1x
m N/2 e mx
p
⇡
detC
where v > = (e m(tN t1 ) ...e m(tN tN
pression for the inverse:
0
(C 1 v)(C 1 v)>
CN1 1 + N 1 µ N 1
1
C ⌘@
1
>
µ
which can be thought of as the Hamiltonian for the simple harmonic oscillator with position coordinate scaled to
have unit mass, and frequency ! = m. The spectrum is
mZ+ and the ground state is given by:
⌦0 (x) =
I can now find an expression for general N using induction. Writing CN in blocks:
◆
✓
CN 1 v
CN ⌘
(8)
v>
1
(CN1 1 v)j = e
m(tN
tN
1)
j,N
(10)
1
which gives:
µ = 1 e 2m(tN tN 1 )
1
1 2
= x>
N 1 CN 1 xN 1 + µ [xN + e
m(tN tN 1 )
2xN xN 1 e
]
1
x>
N CN x N
2m(tN
tN
1)
x2N
(11)
In terms of ⇢N
⇢N =
m
⇡
= ⇢N
= ⇢N
1
2
1,
m
⇡
one thus finds:
N
1 ⇢tN ,tN
1 ⌦0 (xN
1
2
e
mx>
N
p
1 CN
x
1 N
detCN
(xN , xN
1
B tN
1)
1
1
1
1 )⌦0 (xN
tN
e
(x> C
m
p
1
1
(xN
1)
1 x)
µ
2
1 , xN )⌦0 (xN )
(12)
The expressions for N = 1 and N = 2 are both consistent
with the following expression for general N :
⇢N = ⌦0 (x1 )Bt2 t1 (x1 , x2 )Bt3 t2 (x2 , x3 )...
...BtN tN 1 (xN 1 , xN )⌦0 (xN )
where t1 < ... < tN .
(13)
1

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