A seminar on "Hilbert space representation of the minimal length uncertainty relation" paper. This paper is about quantum gravity phenomenology.

1. Quantum Gravity
Phenomenology
Hilbert Space Representation of the Minimal Length Uncertainty
Relation
May 17, 2018

2. Content List
 Reminding
 Introduction
 Minimal length uncertainty
relations
 Hilbert space representation
 Representation theoretic
consequences of the uncertainty
relations
 Representation on momentum
space
 Functional analysis of the position
operator
 Recovering information on
position
 Maximal localization states
 Transformation to quasi-position
wave functions
 Quasi-position representation
 Generalization to n dimensions
Generalized Heisenberg algebra
for n dimensions
Outlook
2

3. Reminding
3

4. Reminding
4

5. Reminding
Some Proposals on the GUP
▪ KMM
▪ ADV
▪ Maggiore
▪ String Theory
▪ Ng and Van Dam
▪ ∆𝑥 𝑚𝑖𝑛 = 3𝛽0 𝑙 𝑝
▪ ∆𝑥 𝑚𝑖𝑛 = 2𝜂0 𝑙 𝑝
▪ ∆𝑥 𝑚𝑖𝑛 = 𝛾0
2 𝑙 𝑝
▪ ∆𝑥 𝑚𝑖𝑛 = 𝛽0 𝑙 𝑝
▪ ∆𝑙 𝑡𝑜𝑡 𝑚𝑖𝑛 = 3 𝑙 𝑝
2
𝑙
1
3
5

6. And Now
▪ In this seminar, we shall discuss
KMM proposal in full detail.
▪ So, this is a detailed review of the
journal paper, Hilbert Space
Representation of the Minimal
Length Uncertainty Relation, by
Achim Kempf, Gianpiero Mangano,
and Robert B. Mann, (1996).
▪ In this context a generalized
uncertainty relation has been
derived which quantum theoretically
describes the minimal length as a
minimal uncertainty in position
measurements.
6

7. Introduction
7

8. Introduction
Quantum gravity:
▪ Introduction of gravity into
quantum field theory appears to
spoil their renormalizability.
▪ Gravity: An effective cutoff in the
ultraviolet, i.e. to a minimal
observable length.
▪ The high energies used in trying to
resolve small distances will
eventually significantly disturb the
spacetime structure by their
gravitational effects.
8
• Renormalization is a collection of
techniques that are used to treat
infinities arising in calculated
quantities by altering values of
quantities to compensate for
effects of their self-interactions.

9. Introduction
Quantum gravity:
▪ If indeed gravity induces a lower
bound to the possible resolution of
distances, gravity could in fact be
expected to regularize quantum
field theories rather than rendering
them nonrenormalizable.
9

10. Introduction
▪ The purpose: Developing a
generalized non-relativistic
quantum theoretical framework
which implements the appearance
of a nonzero minimal uncertainty in
positions.
▪ The more general case includes
nonzero minimal uncertainties in
momenta as well as position and
we must use a generalized
Bargmann-Fock space
representation.
▪ In this paper, we have only the
minimal uncertainty in position and
taking the minimal uncertainty in
momentum to vanish. So there still
exists a continuous momentum
space representation.
10

11. Minimal length uncertainty
relations
11

12. Minimal length uncertainty relations
▪ The simplest generalized
uncertainty relation which leads to
a nonzero minimal uncertainty ∆𝒙 𝟎
in position:
∆𝑥∆𝑝 ≥
ℏ
2
1 + 𝛽 Δ𝑝 2 + 𝛾
⇒ ∆𝑥0≥
ℏ 2𝛽
4
3 + 𝛾
∆𝐴∆𝐵 ≥
ℏ
2
𝐴, 𝐵
𝑥, 𝑝 = 𝑖ℏ 1 + 𝛽𝑝2
, 𝛾 = 𝛽 𝑝 2
12

13. Minimal length uncertainty relations
▪ The more general case which
leads to a nonzero minimal
uncertainty in both position ∆𝑥0
and momentum ∆𝑝0:
∆𝑥∆𝑝 ≥
ℏ
2
1 + 𝛼 ∆𝑥 2 + 𝛽 Δ𝑝 2 + 𝛾
∆𝐴∆𝐵 ≥
ℏ
2
𝐴, 𝐵
𝑥, 𝑝 = 𝑖ℏ 1 + 𝛼𝑥2 + 𝛽𝑝2
𝛾 = 𝛼 𝑥 2
+ 𝛽 𝑝 2
▪ This general case is far more
difficult to handle, since neither a
position nor a momentum space
representation is viable. Instead
one has to resort to a generalized
Bargmann-Fock space
representation.
13

14. Hilbert space representation
14

15. Hilbert space representation (of such a commutation relation)
▪ We generally require physical
states:
– to be normalizable
– to have well-defined expectation
values of position and momentum
– to have well-defined uncertainties
in position and momentum
▪ This implies that physical states
always lie in the common domain
𝐷 𝑥,𝑥2,𝑝,𝑝2 of the symmetric
operators 𝑥, 𝑥2, 𝑝, 𝑝2.
15

16. Representation theoretic consequences of
the uncertainty relations
▪ In ordinary quantum mechanics:
𝜓 𝑥 ≔ 𝑥 𝜓
𝜓 𝑝 ≔ 𝑝 𝜓
𝑥 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑒𝑖𝑔𝑒𝑛𝑠𝑡𝑎𝑡𝑒
𝑝 𝑖𝑠 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑒𝑖𝑔𝑒𝑛𝑠𝑡𝑎𝑡𝑒
▪ Strictly speaking the 𝑥 and 𝑝
are not physical states since they
are not normalizable and thus not
in the Hilbert space.
▪ 𝑥 and 𝑝 are essentially self-adjoint.
▪ These eigenstates can be
approximated to arbitrary precision
by sequences 𝜓 𝑛 of physical
states of increasing localization in
position or momentum space:
lim
𝑛→∞
∆𝑥 𝜓 𝑛
= 0
lim
𝑛→∞
∆𝑝 𝜓 𝑛
= 0
16

17. Representation theoretic consequences of
the uncertainty relations
▪ This situation changes When we
have the minimal uncertainties
∆𝑥0≥ 0 and/or ∆𝑝0≥ 0.
▪ A nonzero minimal uncertainty in
position implies that there cannot
be any physical state which is a
position eigenstate since an
eigenstate would have zero
uncertainty in position:
𝜓 ∆𝑥 2 𝜓 ≡ ∆𝑥 𝜓
2
=
𝜓 𝑥 − 𝜓 𝑥 𝜓 2
𝜓 ≥ ∆𝑥0 ∀ 𝜓
17
▪ Of course, we still have unphysical,
‘formal position eigenvectors’ which
lie in the domain of x alone but not in
𝐷 𝑥,𝑥2,𝑝,𝑝2.
▪ Unlike in ordinary quantum
mechanics, it is no longer possible to
approximate formal eigenvectors
through a sequence of physical
states of uncertainty in positions
decreasing to zero, because now all
physical states have at least a finite
minimal uncertainty in position.
▪ A minimal uncertainty in position
will mean that the position operator
is no longer essentially self-adjoint
but only symmetric.
▪ Since there are then no more
position eigenstates 𝑥 in the
representation of the Heisenberg
algebra, the Heisenberg algebra
will no longer find a Hilbert space
representation on position wave
function 𝑥 𝜓 .

18. Representation on momentum space
∆𝑥∆𝑝 ≥
ℏ
2
1 + 𝛽 Δ𝑝 2
+ 𝛽 𝑝 2
𝑥, 𝑝 = 𝑖ℏ 1 + 𝛽𝑝2
∆𝑥 𝑚𝑖𝑛 𝑝 = ℏ 𝛽 1 + 𝛽 𝑝 2
▪ The absolutely smallest
uncertainty in positions which
occurs when 𝑝 = 0 is:
∆𝑥0= ℏ 𝛽
∆𝑝 =
∆𝑥
ℏ𝛽
±
∆𝑥
ℏ𝛽
2
−
1
𝛽
− 𝑝 2
18
▪ There is no nonvanishing minimal
uncertainty in momentum. In fact
the Heisenberg algebra can be
represented on momentum space
wave functions 𝜓 𝑝 ≔ 𝑝 𝜓 .

19. Representation on momentum space
𝑝. 𝜓 𝑝 = 𝑝𝜓 𝑝
𝑥. 𝜓 𝑝 = 𝑖ℏ 1 + 𝛽𝑝2 𝜕 𝑝 𝜓 𝑝
⇒ 𝑥, 𝑝 = 𝑖ℏ 1 + 𝛽𝑝2
▪ 𝑥 and 𝑝 are symmetric:
𝜓 𝒑 𝜙 = 𝜓 𝒑 𝜙
𝜓 𝒙 𝜙 = 𝜓 𝒙 𝜙
▪ But now, by the following scalar
product the symmetry of 𝑝 is
obvious:
𝜓 𝜙 =
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝜓∗ 𝑝 𝜙 𝑝
▪ The symmetry of 𝑥 can be seen by
performing a partial integration:
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝜓∗ 𝑝 𝑖ℏ 1 + 𝛽𝑝2 𝜕 𝑝 𝜙 𝑝
=
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝑖ℏ 1 + 𝛽𝑝2 𝜕 𝑝 𝜓 𝑝
∗
𝜙 𝑝
▪ Thereby the 1 + 𝛽𝑝2 −1 -factor of the
measure on momentum space is
needed to cancel a corresponding
factor of the operator representation of
𝑥.
19
▪ The identity operator:
1 =
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝑝 𝑝
▪ The scalar product of momentum
eigenstates:
𝑝 𝑝′ = 1 + 𝛽𝑝2 𝛿 𝑝 − 𝑝′
▪ The momentum operator still is
essentially self-adjoint.

20. Functional analysis of the position
operator
▪ So we obtain formal position
eigenvectors which are not physical
states because of the uncertainty
relation:
𝜓 𝜆 𝑝 =
𝛽
𝜋
𝑒𝑥𝑝 −𝑖
𝜆
ℏ 𝛽
tan−1
𝛽𝑝
▪ The scalar product of the formal
position eigenstates:
𝜓 𝜆′ 𝜓 𝜆
=
𝛽
𝜋 −∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝑒𝑥𝑝 −𝑖
𝜆 − 𝜆′
ℏ 𝛽
tan−1
𝛽𝑝
=
2ℏ 𝛽
𝜋 𝜆 − 𝜆′
sin
𝜆 − 𝜆′
2ℏ 𝛽
𝜋
20
▪ The eigenvalue problem for the
position operator on momentum
space:
𝑖ℏ 1 + 𝛽𝑝2 𝜕 𝑝 𝜓 𝜆 𝑝 = 𝜆𝜓 𝜆 𝑝
𝜓 𝜆 𝑝 = 𝑐𝑒𝑥𝑝 −𝑖
𝜆
ℏ 𝛽
tan−1
𝛽𝑝
𝜆 = ±𝑖
1 =
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝜓 𝜆 𝑝 𝑝 𝜓 𝜆
1 = 𝑐𝑐∗
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
= 𝑐𝑐∗
𝜋
𝛽

21. Functional analysis of the position
operator
21

22. Functional analysis of the position
operator
▪ The formal position eigenstates
are generally no longer orthogonal.
▪ The sets of eigenvectors
parametrized by 𝜆 ∈ −1,1 consist
of mutually orthogonal
eigenvectors:
𝜓 2𝑛+𝜆 ℏ 𝛽 𝑛 ∈ ℤ
𝜓 2𝑛+𝜆 ℏ 𝛽 𝜓 2𝑛′+𝜆 ℏ 𝛽 = 𝛿 𝑛,𝑛′
▪ Each of these lattices of formal 𝑥-
eigenvectors has the lattice
spacing 2ℏ 𝛽 which is also 2∆𝑥0.
▪ We are now describing physics on
lattices in position space.
▪ This is however not the case since
the formal position eigenvectors
𝜓 are not physical states. This is
because:
22

23. Functional analysis of the position
operator
▪ The formal position eigenvectors
are not in the domain of 𝑝 (but lie
in the domain of 𝑥 only) which
physically means that they have
infinite uncertainty in momentum
and in particular also infinite
energy:
𝜓 𝜆
𝒑2
2𝑚
𝜓 𝜆 = 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡
▪ Vectors 𝜓 that have a well
defined uncertainty in position ∆𝑥 𝜓
which is inside the forbidden gap
0 ≤ ∆𝑥 𝜓< ∆𝑥0 can not have finite
energy.
23

24. Recovering information on
position
24

25. Recovering information on position
▪ Generally in quantum mechanics
all information on position is
encoded in the matrix elements of
the position operator.
▪ Matrix elements can of course be
calculated in any basis, e.g. also in
the momentum eigenbasis.
▪ We now no longer have any
position eigenbasis of physical
states 𝑥 whose matrix elements
𝑥 𝜓 would have the usual direct
physical interpretation about
positions.
▪ Nevertheless all information on
position is of course still
accessible.
25

26. Maximal localization states
▪ The states 𝜓 𝜉
𝑚𝑙
of maximal
localization around a position 𝜉:
𝜓 𝜉
𝑚𝑙
𝒙 𝜓 𝜉
𝑚𝑙
= 𝜉
Δ𝑥
𝜓 𝜉
𝑚𝑙 = Δ𝑥0
▪ For each state in the representation
of the Heisenberg algebra we
deduce:
𝜓 𝑥 − 𝑥 2
−
𝑥, 𝑝
2 Δ𝑝 2
2
𝑝 − 𝑝 2
𝜓
≥ 0
26
∆𝑥∆𝑝 ≥
𝑥, 𝑝
2
▪ A state 𝜓 will obey ∆𝑥∆𝑝 =
𝑥,𝑝
2
i.e. it is on the boundary of the
physically allowed region only if it
obeys:
𝑥 − 𝑥 +
𝑥, 𝑝
2 Δ𝑝 2
𝑝 − 𝑝 𝜓 = 0

27. Maximal localization states
▪ In momentum space:
𝑖ℏ 1 + 𝛽𝑝2
𝜕 𝑝 − 𝑥 + 𝑖ℏ
1 + 𝛽 Δ𝑝 2
+ 𝛽 𝑝 2
2 Δ𝑝 2
𝑝 − 𝑝 𝜓 𝑝 = 0
𝜓 𝑝 = 𝑁 1 + 𝛽𝑝2 −
1+𝛽 Δ𝑝 2+𝛽 𝑝 2
4𝛽 Δ𝑝 2
𝑒𝑥𝑝
𝑥
𝑖ℏ 𝛽
−
1 + 𝛽 Δ𝑝 2
+ 𝛽 𝑝 2
𝑝
2 Δ𝑝 2 𝛽
tan−1
𝛽𝑝
▪ The states of absolutely maximal localization can only be obtained for 𝑝 = 0.
Choose the critical momentum uncertainty Δ𝑝 = 1/ 𝛽:
𝜓 𝜉
𝑚𝑙
𝑝 = 𝑁 1 + 𝛽𝑝2 −
1
2 𝑒𝑥𝑝 −𝑖
𝑥 tan−1
𝛽𝑝
ℏ 𝛽
, 𝑁 =
2 𝛽
𝜋
, 𝑥 = 𝜉
𝜓 𝜉
𝑚𝑙
𝑝 =
2 𝛽
𝜋 1 + 𝛽𝑝2 −
1
2 𝑒𝑥𝑝 −𝑖
𝜉 tan−1
𝛽𝑝
ℏ 𝛽
27
• These states generalize the plane waves in momentum space or Dirac 𝛿-
‘functions’ in position space which would describe maximal localization in
ordinary quantum mechanics.
• Unlike the latter, the new maximal localization states are now proper physical
states of finite energy:
𝜓 𝜉
𝑚𝑙 𝑝2
2𝑚
𝜓 𝜉
𝑚𝑙
=
2 𝛽
𝜋
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2 2
𝑝2
2𝑚
=
1
2𝑚𝛽

28. Maximal localization states
▪ Due to the fuzziness of space the maximal localization states are in general
no longer mutually orthogonal:
𝜓 𝜉′
𝑚𝑙
𝜓 𝜉
𝑚𝑙
=
2 𝛽
𝜋
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2 2
𝑒𝑥𝑝 −𝑖
𝜉 − 𝜉′
tan−1
𝛽𝑝
ℏ 𝛽
𝑝 = tan−1
𝛽𝑝 , 𝑑 𝑝 =
𝛽𝑑𝑝
1 + 𝛽𝑝2
sin2 tan−1 𝛽𝑝 =
𝛽𝑝2
1 + 𝛽𝑝2
, cos2 tan−1 𝛽𝑝 =
1
1 + 𝛽𝑝2
𝜓 𝜉′
𝑚𝑙
𝜓 𝜉
𝑚𝑙
=
1
𝜋
𝜉 − 𝜉′
2ℏ 𝛽
−
𝜉 − 𝜉′
2ℏ 𝛽
3
−1
sin
𝜉 − 𝜉′
2ℏ 𝛽
𝜋
28

29. Maximal localization states
29

30. Transformation to quasi-position wave
functions
▪ While in ordinary quantum
mechanics it is often useful to
expand the states 𝜓 in the
position eigenbasis { 𝑥 } as 𝑥 𝜓 ,
there are now no physical states
which would form a position
eigenbasis.
▪ We can still project arbitrary states
𝜙 on maximally localized states
𝜓 𝜉
𝑚𝑙
to obtain the probability
amplitude for the particle being
maximally localized around the
position 𝜉.
▪ We call the collection of these
projections 𝜓 𝜉
𝑚𝑙
𝜙 the state’s
quasi-position wavefunction 𝜙 𝜉 :
𝜙 𝜉 ≔ 𝜓 𝜉
𝑚𝑙
𝜙
30

31. Transformation to quasi-position wave
functions
▪ The transformation of a state’s wavefunction in the momentum
representation into its quasi-position wavefunction:
𝜓 𝜉 =
2 𝛽
𝜋
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
3
2
𝑒𝑥𝑝 𝑖
𝜉 tan−1
𝛽𝑝
ℏ 𝛽
𝜓 𝑝
▪ The transformation of a quasi-position wavefunction into a momentum space
wavefunction:
𝜓 𝑝 =
1
8𝜋 𝛽ℏ −∞
+∞
𝑑𝜉 1 + 𝛽𝑝2
1
2 𝑒𝑥𝑝 −𝑖
𝜉 tan−1 𝛽𝑝
ℏ 𝛽
𝜓 𝜉
31

32. Quasi-position representation
▪ The scalar product of states in terms of the quasi-position wavefunctions:
𝜓 𝜙 =
−∞
+∞
𝑑𝑝
1 + 𝛽𝑝2
𝜓∗ 𝑝 𝜙 𝜙
= 8𝜋 𝛽ℏ
−1
−∞
+∞
−∞
+∞
−∞
+∞
𝑑𝑝𝑑𝜉𝑑𝜉′ 𝑒𝑥𝑝 𝑖
𝜉 − 𝜉′
tan−1
𝛽𝑝
ℏ 𝛽
𝜓∗ 𝜉 𝜙 𝜉′
▪ The action of momentum and position operator on quasi-position
wavefunctions:
𝒑. 𝜓 𝜉 =
tan −𝑖ℏ 𝛽𝜕 𝜉
𝛽
𝜓 𝜉
𝒙. 𝜓 𝜉 = 𝜉 + 𝛽
tan −𝑖ℏ 𝛽𝜕 𝜉
𝛽
𝜓 𝜉
32

33. Generalization to n
dimensions
33

34. Generalized Heisenberg algebra for n
dimensions
▪ The aim is to study an n
dimensional generalization of the
framework.
𝒙𝑖, 𝒑 𝑗 = 𝑖ℏ𝛿𝑖𝑗 1 + 𝛽𝒑2
▪ We require 𝒑𝑖, 𝒑 𝑗 = 0 which
allows us to straightforwardly
generalize the momentum space
representation of the previous
sections to n dimensions:
𝒑𝑖. 𝜓 𝜙 = 𝑝𝑖 𝜓 𝜙
𝒙𝑖. 𝜓 𝜙 = 𝑖ℏ 1 + 𝛽𝒑2 𝜕 𝑝 𝑖
𝜓 𝜙
𝒙𝑖, 𝒙𝑗 = 2𝑖ℏ𝛽 𝒑𝑖 𝒙𝑗 − 𝒑 𝑗 𝒙𝑖
▪ The scalar product:
𝜓 𝜙 =
−∞
+∞
𝑑 𝑛 𝑝
1 + 𝛽 𝑝2
𝜓∗ 𝑝 𝜙 𝜙
▪ The identity operator:
1 =
−∞
+∞
𝑑 𝑛 𝑝
1 + 𝛽 𝑝2
𝑝 𝑝
▪ The momentum operators are still
essentially self-adjoint, the position
operators are symmetric and do
not have physical eigenstates.
34

35. Outlook
35

36. Outlook
▪ We have shown in the simplest
non-trivial case that it is no longer
possible to spatially localize a
wavefunction to arbitrary precision.
▪ A minimal length of the form of a
minimal position uncertainty may
indeed describe a genuine feature
of spacetime, arising directly from
gravity.
▪ The quasiposition representation,
should now allow a much more
detailed examination of the
phenomenon of a minimal position
uncertainty in quantum field theory
than has been possible so far.
▪ A quantum theory with minimal
uncertainties might be useful as an
effective theory of nonpointlike
particles.
36

37. References
37
▪ Kempf, A., Mangano, G., & Mann, R. B. (1996). Hilbert Space
Representation of the Minimal Length Uncertainty Relation. PHYSICAL
REVIEW D.

38. 38
The robotic
rover Spirit,
2005

39. 39
NASA's
Curiosity
Mars rover,
2014,
about 80
minutes
after sunset