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Stars Overhead – Lorre On-Screen
In this summer of 2012, several cities across the United States are offering outdoor showings of classic Hollywood movies. Here are just a few with Peter Lorre films on the schedule.
Syracuse, New York:
The Everson Museum of Art in Syracuse, NY, will present their annual "Film Under the Stars" summer series in partnership with On-Center and the Urban Video Project. On the schedule this season are Casablanca (1942) and 20,000 Leagues Under the Sea (1954).
Friday, July 27, 2012, is "Date Night", with Casablanca beginning at 8:30 pm.
Friday, Aug. 10 is "Sci-Fi Cult Classics Night", with 20,000 Leagues Under the Sea beginning at 8:30 pm.
All films are shown in the outdoor plaza. Admission is free, and the public is invited to bring lawn chairs and blankets, as well as food and beverages to enjoy during the screening.
In case of rain, the start time will be delayed one hour. However, should the event be rained out entirely that night, the movie will be shown the following day at the same time. Any notice of rescheduling will be posted on the Everson Museum home page.
For more information, please call the museum at (301) 474-6064.
The Everson Museum of art is located at 401 Harrison Street, in Syracuse, NY. Free parking is available at the open lot on the corner of Harrison and State streets, directly across from the Museum.
San Diego, California:
Beat the Devil, Peter Lorre's final movie with on- and off-screen pal Humphrey Bogart, will be shown on Saturday, July 28, as part of the San Diego Museum of Art's annual "Screen on the Green" series.
The movie is presented in conjunction with The Invention of Glory, a display of the Pastrana Tapestries from now until Sept. 9, 2012. According to the Museum, the tapestries "depict European traveling to Africa to conquer lands and control resources" – though not in the comic fashion of the movie's gallery of rogues.
All films will be shown on the Museum's East Lawn, in the heart of Balboa Park, beginning at sundown, approximately 8 pm. Admission is free, and moviegoers are invited to bring their own lawn chairs, blankets, and picnic baskets to the screenings.
The San Diego Museum of Art is located in the heart of Balboa Park in San Diego, CA, at 1450 El Prado. Parking is available throughout the park, and the closest lot is the Plaza de Panama, located next to the Museum. Valet parking is also offered in front of the House of Hospitality in the Plaza de Panama.
The Museum's website includes directions to SDMA, as well as more information on parking in the area. For more information, please call (619) 232-7931.
Albany, Oregon:
On Friday, Aug. 3, the Albany Downtown Association will present My Favorite Brunette (1947) as part of their annual "Movies by Moonlight" series.
Admission is free, and the program begins with seating at 7 pm, followed at dusk by a classic cartoon and an episode of a Superman serial preceding the main feature. Popcorn, ice cream, water, and sodas will be available for purchase.
All movies are shown in the parking lot behind Cappie's Brewhouse and projected against the back wall of the Eagles Club next door. Cappie's is located at 211 First Avenue West, in Albany, Oregon.
Pizza King, the Pix Theatre, Cappie's, and Pacific Power jointly sponsor the "Movies by Moonlight" series. For more information, please call (541) 928-2469.
In the pages of the authorized Lorre biography, The Lost One: A Life of Peter Lorre, author Stephen Youngkin recounts Peter's experiences working on all these movies, including interviews with directors John Huston and Richard Fleischer, screenwriter Julius J. Epstein, and more. The Lost One is now available in paperback, as well as hard-bound and the Kindle and Nook.
Posted by Cheryl Morris 0 comments Labels: 20000 Leagues Under the Sea, Albany Downtown Association, Beat the Devil, Casablanca, Everson Museum of Art, My Favorite Brunette, San Diego Museum of Art
The Raven Under the Stars in Camden, Maine
Peter Lorre fans in Camden, Maine, will have an opportunity to enjoy an outdoor showing of The Raven (1963) when the Camden Public Library begins its summer series "Movies in the Amphitheater" on Tuesday, July 17, 2012.
Admission is free, and the public is invited to bring lawn chairs and blankets, in addition to drinks and snacks. The library will serve cheese puffs – as a tribute to the "cheesy" special effects featured in the film, particularly in the final duel between magicians Vincent Price and Boris Karloff.
The program begins after sundown at 8:15 pm. In case of rain, The Raven will move indoors to the Picker Room in the library.
The Camden Public Library is located at 55 Main Street, in downtown Camden, Maine. Directions may be found on the library's website. The Amphitheatre is located beside the library, along Atlantic Avenue. Parking is available on the streets near the library.
For more information, please contact the library at (207) 236-3440.
In the summer of 1974, Roger Corman reminisced with Lorre biographer Stephen Youngkin on directing Peter Lorre in The Raven, as well as Tales of Terror (1962). In failing health and unable to remember his lines, Peter preferred to ad-lib – to the frustration of the stage-trained Boris Karloff.
The three actors would work together only once more – in 1964's The Comedy of Terrors – before Peter's passing on March 23, 1964.
Posted by Cheryl Morris 0 comments Labels: Camden Public Library, The Raven
Secret Agent at the AFI in Silver Spring, MD
The American Film Institute theater in Silver Spring, MD, presents Spy Cinema, a series of espionage films from June 29 to Sept. 19, 2012 – and included in the retrospective is the Alfred Hitchcock classic Secret Agent (1936).
Show times for Secret Agent:
July 14, Saturday – 6:30 pm
July 15, Sunday – 11 am
July 16, Monday – 7 pm
Tickets are $11.50 for general admission; $9 for seniors 65 years and older, students, and the military (with proper ID); and $7 for children 12 and under. For AFI members, admission is $8.50. AFI Member Passes will be accepted for all movies in the Spy Cinema series.
Tickets for may be reserved on-line through the AFI Silver website. Click the "Buy Tickets" button, then scroll to the title Secret Agent, click the link and select a show date and time. Tickets reserved on-line must be claimed at the box office with the same credit card used to make the reservation. The AFI Silver accepts American Express, Visa, MasterCard, and Discover credit cards.
Tickets may also be purchased at the AFI Silver box office, which opens 30 minutes before the first film of the day.
The AFI Silver Theatre is located at 8633 Colesville Road, at the intersection of Colesville Road and Georgia Avenue, in Silver Spring, MD. Directions to the cinema may be found on the AFI Silver website.
Parking is available in several public parking decks close to the theater. Most convenient is the Wayne Avenue garage, located at 921 Wayne Avenue, Silver Spring, MD. Parking at the Wayne Avenue garage is free on weekends and after 8 pm on weekdays. More information on parking can be found on the AFI Silver website, under Parking.
For his book The Lost One: A Life of Peter Lorre, author Stephen D. Youngkin interviewed Alfred Hitchcock on Oct. 17, 1973, as the director recalled working with Peter on Secret Agent and The Man Who Knew Too Much (1934).
But while The Man Who Knew Too Much marked the start of Peter's screen association with Hitchcock,Secret Agent essentially ended it.
Towards the end of his career, Peter Lorre appeared in two episodes of the television series Alfred Hitchcock Presents: "The Diplomatic Corpse" (Dec. 8, 1957) and "Man From the South" (Mar. 13, 1960). The latter episode was postponed from its original air-date of Jan. 3, 1960, as CBS considered its gruesome storyline inappropriate for the holiday season. Although Hitchcock hosted the episodes, neither was among those he himself directed.
Posted by Cheryl Morris 0 comments Labels: AFI Silver Theatre, Secret Agent
Peter Lorre on TV This Summer
July, August, and September bring many Peter Lorre movies on Turner Classic Movies. All times shown are Eastern Standard.
Peter Lorre's page on the TV Guide Channel website lists the Lorre films scheduled on various television channels over a 2-week period.
July 7 (Sat.), 11:30 pm, Antenna TV – Alfred Hitchcock Presents, "Man From the South", originally broadcast Mar. 13, 1960.
July 8 (Sun.), 5:45 am, Turner Classic Movies channel – Around the World in 80 Days (1956).
July 10 (Tues.), 4 am, TCM – Confidential Agent (1945).
July 17 (Tues.), 4:30 am, TCM – The Beast With Five Fingers (1946).
July 24 (Tues.), 1:15 pm, TCM – Around the World in 80 Days (1956).
August brings TCM's annual "Summer Under the Stars" festival, with each day devoted to a different movie star.
Aug. 29 (Wed.), 10 pm, TCM – Casablanca (1942). For "Ingrid Bergman Day", TCM presents a 70th anniversary showing of Casablanca.
Sept. 2 (Sun.) – It's "Keep Your Hands to Yourself" night on "The Essentials", and TCM celebrates with two "handy" Lorre movies: The Beast With Five Fingers (1946) at 9:45 pm and Mad Love (1935) at 11:30 pm.
Sept. 12 (Wed.), 9 pm, TCM – Confidential Agent (1945). September's "Star of the Month" Lauren Bacall appears in her only Lorre movie, filmed shortly after her marriage to Humphrey Bogart – and Peter's marriage to his second wife Kaaren Verne.
Sept. 12 (Wed.), 4:15 am, TCM – Passage to Marseille (1944). Bogart was at work on this movie when Lauren Bacall was brought to the set to meet him, prior to their first picture together, To Have and Have Not. It was also the last time Bogie would work with his pal Peter Lorre while Peter was under contract to Warner Bros.
Sept. 28 (Fri.), 4:30 pm, TCM – Silk Stockings (1957).
Many of these and other Lorre movies are now available on DVD and VHS – some remastered and packaged with extra features. For more information on the films of Peter Lorre released to home video, head to the DVD — VHS section of The Lost One website.
Posted by Cheryl Morris 0 comments Labels: Movies on TV 2012
Casablanca On-Screen This Summer
This summer, Peter Lorre fans in the U.S. will have many opportunities to catch Casablanca (1942) in a variety of venues, all in celebration of the film's 70th anniversary.
Elizabeth, North Carolina
The Brattle Theatre in Cambridge, MA, presents Casablanca over the 4th of July holiday. The movie is a staple on the Brattle calendar, and according to the cinema's website, has been shown so many times over the past 50 years, the audience exhibits "cult-like behavior" at presentations – including wedding proposals in the balcony.
Casablanca will be shown July 4 through July 8, Wednesday through Sunday, with matinees at 2 pm and 4:30 pm, as well as an evening show at 7 pm.
Ticket prices for general admission are $7.75 for the matinee and $9.75 for the evening show. For students and Brattle members, tickets are $7.75 for all shows. Admission for senior citizens is $6.75 for all shows. More information on tickets may be found on the Brattle's website.
Tickets may be purchased over the Brattle website. Brattle Passes will be accepted for this engagement.
The Brattle Theatre is located at 40 Brattle Street inside Brattle Hall, one block from Howard Square, in Cambridge, MA. Sharing the building are the Algiers Café, Casablanca Restaurant, and Harvard Square Optical. The cinema's entrance is on the left side of the building, marked by a sidewalk poster case and marquee.
Directions to the Brattle and information about parking may be found on the theater's website.
Parking is available on the street around Harvard Square. The Brattle will also validate parking tickets from these participating garages – the Charles Square Garage, located at the Charles Hotel, 5 Bennett Street; and the University Place Garage, located at 124 Mt. Auburn Street. The Brattle may also reached by public bus and subway.
For more information, please call the cinema at (617) 876-6838.
On Tuesday, July 10, Lorre fans in Elizabeth City, NJ, can experience Casablanca under the stars – on a large screen with quality surround sound audio – as part of the Mariners' Wharf Film Festival.
The program will begin at 8:30 pm – but in case of inclement weather, the screening will be cancelled and rescheduled at a later date.
Admission is free. Bring lawn chairs or blankets and a picnic dinner.
Mariners' Wharf is located at South Water Street, along the Pasquotank River in downtown Elizabeth City, N. Carolina. Parking is available at the wharf and along S. Water Street.
For more information, please contact the festival's founder and producer Simone Cooper at (646) 567-6951 or through her website.
Casablanca is also on the schedule for the Summer Classic Movies Series, shown Wednesdays at the historic Kentucky Theater in Lexington, KY. Although many cinemas are transitioning to digital projection, the Kentucky has been able to acquire a 35mm print of Casablanca, as well as the other movies in the festival.
Casablanca will be shown on Wednesday, July 18, at 1:30 pm and 7:15 pm.
Admission is $5 for all seats.
The Kentucky Theater is located at 214 East Main Street, in Lexington, KY. Directions may be found on the theater's website.
Parking is available in the Annex Garage, at 150-160 West Main Street between Martin Luther King Jr. Blvd. and South Limestone, beside the Fayette County Clerk's office. Hours are Monday through Friday, 7:30 am to 6 pm. After 6 pm, parking is free.
For more information, please call the Kentucky Theater at (859) 231-7924.
On Friday, July 20, the Aero Theatre in Santa Monica, CA, will present Casablanca with another Bogart classic The African Queen (1951) as part of their series of Classics and Restorations on the Big Screen.
The program begins with Casablanca at 7:30 pm.
Ticket prices are $11 for general admission, $7 for Cinematheque Members, and $9 for students with valid ID and senior citizens 65 and over. Tickets may be purchased on-line through Fandango and the Aero website.
The Aero Theatre is located at 1328 Montana Avenue, at 14th Street, in Santa Monica, CA. Parking is available free and at meters on Montana Avenue and other streets around the Aero. Please check posted signs. North of Montana Avenue, parking is generally limited to two hours until 7 pm. After 7 pm, parking is free for the duration of a movie.
For more information, please call the Aero at (310) 260-1528.
The Tampa Theatre is the venue for Casablanca on Aug. 18 and 19, Saturday and Sunday, the final film in that cinema's popular Summer Classics series, running this year from June 3 to Aug. 19, 2012.
The doors open at 2 pm, at which time interested moviegoers may join a volunteer "Tampa Theatreologist" for a free guided mini-tour of the historic theater.
A 20-minute musical program of popular tunes on the cinema's restored Wurlitzer Theatre Organ will begin at approximately 2:40, followed by Looney Tunes cartoons and the movie at 3 pm.
Ticket prices for Casablanca and other Summer Classic films are $9 for all ages and $7 for Tampa Theatre members. Tickets may be purchased on-line through the theater's website, as well as at the box-office. For more information, please call (813) 274-8981.
The Tampa Theatre is located at 711 Franklin Street, in Tampa, FL. Parking is available on-street and close to the cinema. Directions to the cinema are posted on the website.
Casablanca – a classic movie celebrating its 70th anniversary this year. Peter Lorre's experiences on-set are described in the pages of his authorized biography, The Lost One: A Life of Peter Lorre, written by Stephen D. Youngkin and available in paperback, as well as the Kindle, Nook, and hard-bound.
Posted by Cheryl Morris 0 comments Labels: Aero Theatre, Brattle Theatre, Casablanca, Kentucky Theater, Mariners' Wharf Film Festival, Tampa Theatre
So how is "Lorre" pronounced, anyway?
I've seen this question posed on any number of web sites. Peter himself pronounced it "Law-ree". Most others say "Lore-ee".
But "Law-ree" or "Lore-ee", he's still the classic film actor of the 1930s, 1940s, 1950s, and early 1960s.
Lorre on Television
Wondering where to find Peter Lorre on American TV channels this month? Check out this post from the Peter Lorre News blog, listing Peter's movies, TV shows, and documentaries ~ Added 2021 January 4 ~
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Cheryl Morris
I love classic films, especially the work of my favorite actor, Peter Lorre. And if you're a Lorre fan, too, I'd love to hear from you! All comments are welcome.
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Baldwin pears are semi-hard, sweet pears great for canning or desserts. The pears are medium-sized and yellow and ripen in early fall. They are moderately blight and spot resistant and the tree is low chill for the South. The Baldwin requires 150 chill hours. LeConte or Orient make good pollinators for Baldwin. At maturity, it can reach a height between 15-20 feet and a width of 12-15 feet. USDA ZONES: Zone 5, Zone 6, Zone 7, Zone 8, Zone 9.
Came on time (only two days) well packaged. Transplanted with no shock.


text:
Q: IP Resource Errors While Building a Windows Failover Cluster on Azure I have setup a failover cluster in Microsoft Azure on four VMs. I have two nodes in one subnet/region, one node in another subnet/region and a final node in a third subnet/region. The cluster passed the validation wizard and was successfully built, but the two nodes in other subnets will not go online, reporting "Failed to bring the resource 'IP Address x.x.x.x' online".

Azure Vnet Setup

*

*Vnet 1 - South Central US - 10.16.16.0/16 - Nodes 1 & 2 - Online

*Vnet 2 - West US - 10.116.16.0/16 - Node 3 - Offline

*Vnet 3 - East US - 10.216.16.0/16 - Node 4 - Offline

All three IP Address resources are DHCP. I've attempted to make them static but they still failed while coming online. If I click "Information Details" on the error I get:
Error code: 0x80071397
The operation failed because either the specified cluster node is not the owner of the resource, or the node is not a possible owner of the resource.
It seems to be an issue with the way Azure DHCP works, but I'm stalled. There must be a way to get his working. What am I missing?

A: only one can be online at a time. if you go to the list of nodes, and turn off the node where its ip is the one being used by the cluster (10.16.16), then another ip address in your list will come online.

redpajama_set_name: RedPajamaStackExchange


text:
#include "audioblocks.h"
#include <iostream>

class GainBlock : public AudioBlock {
public:
    GainBlock(double g){
        // clamp between 0 and 1
        gain = (g > 1.0 ? 1.0 : (g < 0.0 ? 0.0 : g));
    }
    virtual ~GainBlock(){}
    int process(AudioBuffer<float>& inputbuffer, AudioBuffer<float>& outputbuffer){
        outputbuffer.clear();
        for(float sample : inputbuffer){
            outputbuffer.push_back(sample * gain);
        }

        return abContinue;
    }
private:
    double gain;
};

#endif // __AUDIOBLOCK_GAIN_H__

redpajama_set_name: RedPajamaGithub


text:
/* eslint-env jest */
/**
 * @fileoverview Enforce distracting elements are not used.
 * @author Ethan Cohen
 */

// -----------------------------------------------------------------------------
// Requirements
// -----------------------------------------------------------------------------

import { RuleTester } from 'eslint';
import parserOptionsMapper from '../../__util__/parserOptionsMapper';
import rule from '../../../src/rules/no-distracting-elements';

// -----------------------------------------------------------------------------
// Tests
// -----------------------------------------------------------------------------

const ruleTester = new RuleTester();

const expectedError = (element) => ({
  message: `Do not use <${element}> elements as they can create visual accessibility issues and are deprecated.`,
  type: 'JSXOpeningElement',
});

ruleTester.run('no-marquee', rule, {
  valid: [
    { code: '<div />;' },
    { code: '<Marquee />' },
    { code: '<div marquee />' },
    { code: '<Blink />' },
    { code: '<div blink />' },
  ].map(parserOptionsMapper),
  invalid: [
    { code: '<marquee />', errors: [expectedError('marquee')] },
    { code: '<marquee {...props} />', errors: [expectedError('marquee')] },
    { code: '<marquee lang={undefined} />', errors: [expectedError('marquee')] },
    { code: '<blink />', errors: [expectedError('blink')] },
    { code: '<blink {...props} />', errors: [expectedError('blink')] },
    { code: '<blink foo={undefined} />', errors: [expectedError('blink')] },
  ].map(parserOptionsMapper),
});


Works with 400 Series Frenchwood In-Swing Patio Doors from 1988 to Present.
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Controls are situated on the top right hand side of the appliance.


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We refitted all the bedrooms, with new joinery, loose furniture and finishes, with a view to ensuring that the compact rooms were made to feel as spacious as possible.
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Overall, the project brought the hotel up to the brand standards for Hilton Doubletree, completed in phases in only seven months for a budget of £4 million.


text:
Advanced concepts found in Hindu writings
I offer the following extracts from 'Advanced concepts in Hinduism' from the Internet. It is incredible that, way back in time, unknown people in Asia had speculated successfully, or had been told by extra-terrestrials, about matters we consider scientific; and that modern discoveries confirm some of these speculations. The following is an extract from the same site as the material below on the Vedas.
"Advanced Scientific Concepts in Hindu Literature:
Sphericity of Earth, Earth as Flat at poles, Sun the center of the Solar System, Atoms, Universal Time Scale, The Expanding Egg, Concept of Trinity, Hundred thousandths of a second, Airplanes, Description of Tides, Botany and Biology, Electricity and others."
The material below offers a commentary on the Vedas, ancient writings.
"The Vedas have guided Indian civilization for thousands of years. They are the pillars of Hinduism. 'Veda is the source of all Dharma' declares Manusmirti (2.6.) There is no major religion on the planet, which has not been influenced by the Vedas. The creation stories of all major religions are based on Vedas. Though all other religions have forgotten their Vedic root or have been forgotten, there is one religion, Hinduism, that has kept the flame of the Vedic wisdom burning continuously. Vedas which means 'knowledge' contain a good deal of scientific knowledge that was lost over millennia, which needs to be recovered. The Vedic sages had discovered the subtle nature of reality, and had coded it in the form of the Vedas."
"According to Raja Ram Mohan Roy, author of Vedic Physics, "The knowledge contained in the Vedas is very abstruse, and is well beyond the comprehension of ordinary human beings. Therefore Vedic sages coded the knowledge in a simple form in which it could be understood by everyone. The Rig Veda itself testifies that it has a hidden meaning in verse 4.3.16. Sage Bharata in his Natyasastra 2.23 refers to the sages who knew the hidden meaning of the Vedas. This coding of knowledge proved to be very successful in disseminating the knowledge to common folks. This would also explain why extraordinary steps were taken to preserve the Vedas, and the honor given to the Vedas by Hindus, even though its meaning is little understood today. "On the eve of the "Mahabharata War" our ancestors believed that their knowledge was in danger of being lost. They could have written it down, but writings could be destroyed. Therefore, it was memorized and passed on orally. Today, the Avesta, religious scripture of ancient Iranians, only a fraction of it is available. Alexander captured Iran in 326 B.C. and after a bloody war, destroyed each copy of the Avesta available."
"As in modern physics, Hindu cosmology envisaged the universe as having a cyclical nature. The end of each kalpa brought about by Shiva's dance is also the beginning of the next. Rebirth follows destruction."
Author Dick Teresi says "Indian cosmologists, the first to estimate the age of the earth at more than 4 billion years. They came closest to modern ideas of atomism, quantum physics, and other current theories. India developed very early, enduring atomist theories of matter. Possibly Greek atomistic thought was influenced by India, via the Persian civilization."
Apart from the reference to Alexander (who allegedly wondered whether he had been sired by a god), it is fascinating that advanced understanding was available to us so far back in history. Surely it does not matter whether the relevant information was home-grown (by whom or where), or given to us by extra-terrestrials.
Posted in A cosmic mind, A spiritual perspective, Ancient folklore, Asian civilisations, Civilisations, Cosmic laws, Cultural maturity, Extraterrestrials, History, Learning, My Views, Perceiving reality, Religion, Science, The Cosmos | Tagged Did extra-terrestrials provide knowledge to humans?, Hindu literature contains references to advanced scientific concepts, How did early humans conceive complex scientific concepts?, The cyclical nature of Hindu cosmology, The Hindu Vedas apparently pre-date all other writing on Earth, The Vedas contain abstruse knowledge disseminated by sages to common folk |

redpajama_set_name: RedPajamaCommonCrawl


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At the Sorano hot springs, the water comes from more than one spring, spread along a torrent and smaller stream. The water temperature at the source is 37.5°C. They are classified as magnesium-calcium hot springs, as they are rich in magnesium, calcium and zinc. They therapeutic effect has been recognized above all in the fields of dermatology, rheumatology and slight vasculopathy.
Within the structures there are wellness centres, pools and waterfalls.
Agriturismo Poggio Mirabile's clients can take advantage of a 10% discount on the entrance fee.
The Molino waterfalls at Saturnia represent the happy meeting point between wellness and enchantment for the eyes.
The scenographic complex is made of large pools of calcium deposits at different heights formed over the centuries. The hot spring waters of the Gorello torrent jump from one to the other at 37.5°, and decline at the valley.
Access to the waterfalls is free.
Agriturismo Poggio Mirabile has established advantageous agreements with important Spa and wellness centers throughout the Maremma territory.
The hot springs at Sorano are 25 km from Agriturismo Poggio Mirabile.


Orikri's art is profound, expressive, and as with any living artist continually expanding. He adds significantly to the growth, relevance and visibility of contemporary world art, as well as enriches society with his contributions. Beyond his brilliant use of colors and forms, lie the subtleties that will cause you to linger and absorb what you see and feel.


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text:
\section{Introduction}
It is commonly expected that the cosmological singularity (CS)
problem \cite{MTW,SWH,JPAK,JMMS} may be resolved  in a theory
which unifies gravity and quantum physics. It seems that recent
developments concerning quantization of cosmological models  by
making use of  {\it loop geometry} may bring solution to the
problem. It consists in turning the  classical {\it big bang}
singularity into  {\it big bounce}, BB, transition. There exist
two methods to address the issue: {\it standard} loop quantum
cosmology (LQC) and {\it nonstandard} LQC. The former has been
developed during the last decade (see
\cite{Ashtekar:2003hd,Bojowald:2006da} and references therein) and
has been inspired by the loop quantum gravity, LQG (see
\cite{TT,CR,Ashtekar:2004eh} and references therein). The latter
has been proposed recently
\cite{Dzierzak:2008dy,Malkiewicz:2009zd,Dzierzak:2009ip,Dzierzak:2009dj}
and seems to be related to the reduced phase space quantization of
LQG \cite{Giesel:2007wn}.

The standard LQC means basically the Dirac  method of
quantization, which begins with quantization of the {\it
kinematical} phase space followed by imposition of constraints of
the gravitational system in the form of operators at the quantum
level. Finding kernels of these operators helps to define the {\it
physical} Hilbert space. In the nonstandard LQC (our method) one
first solves all the constraints at the classical level to
identify the {\it physical} phase space. Next, one identifies the
algebra of {\it elementary} observables (in the physical phase
space) and finds its representation. Then, {\it composite}
observables are expressed in terms of elementary ones and
quantized. Final goal is finding {\it spectra} of composite
observables which are used to examine the {\it nature} of the BB
phase in the evolution of the universe.

In what follows we restrict our considerations to  the  flat
Friedmann-Robertson-Walker (FRW) model with massless scalar field.
This model of the universe includes the initial cosmological
singularity and has been intensively studied recently within the
standard LQC.

In our recent paper \cite{Dzierzak:2008dy} we have argued  that
the resolution of the singularity offered by LQC requires specific
value of the fundamental {\it length}.  The size of this length
has not been determined satisfactory yet.  Present paper is an
extended version of \cite{Malkiewicz:2009zd} and together with the
classical formalism \cite{Dzierzak:2009ip} specifies our
nonstandard LQC (when applied to FRW model).

In order to have our paper self-contained,  we recall in Sec. II
some aspects of the classical formalism of our nonstandard LQC
\cite{Dzierzak:2009ip}. In Sec. III we present our quantization
procedure. It consists in  finding representation for composite
observables like the energy density and the volume function, and
calculating their spectra. We conclude in the last section.

In the Appendix, we give an extended {\it motivation} for our
paper. In particular, we justify the need for quantization of our
loop FRW model [12], which is free from a cosmological singularity
already at the classical level due to the {\it modification} of
the classical Hamiltonian.

\section{Classical Level}

\subsection{Hamiltonian}

The gravitational part of the classical Hamiltonian, $H_g$, of the
flat FRW model may be presented in the form \cite{Dzierzak:2009ip}
\begin{equation}\label{hamR}
    H_g = \lim_{\mu\rightarrow \,0}\; H^{(\mu)}_g ,
\end{equation}
where
\begin{equation}\label{hamL}
H^{(\mu)}_g = - \frac{sgn(p)}{2\pi G \gamma^3 \mu^3}
\sum_{ijk}\,N\, \varepsilon^{ijk}\, Tr \Big(h^{(\mu)}_i
h^{(\mu)}_j (h^{(\mu)}_i)^{-1} (h^{(\mu)}_j)^{-1}
h_k^{(\mu)}\{(h_k^{(\mu)})^{-1},V\}\Big) ,
\end{equation}
and where $V= |p|^{\frac{3}{2}}= a^3 V_0$ is the volume of the
elementary cell $\mathcal V\subset \Sigma$ ($\Sigma$ is spacelike
hyper-surface); $p$ is a canonical variable; the metric of $k=0$
FRW model is: $ds^2=-N^2(t)\,dt^2+a^2(t)\,(dx^2+dy^2+dz^2),$ where
$a$ is the scale factor and $N$ denotes the lapse function;
$\varepsilon^{ijk}$ is the alternating tensor;  $\gamma$ is the
Barbero-Immirzi parameter, and the holonomy function
$h_k^{(\mu)}$, along straight line of coordinate length
proportional to $\mu/|a|$,  reads
\begin{equation}\label{hol}
h^{(\mu)}_k (\beta) = \cos (\mu \beta/2)\;\mathbb I + 2\,\sin (\mu
\beta/2)\;\tau_k,
\end{equation}
where $\tau_k = -i \sigma_k/2\;$ ($\sigma_k$ are the Pauli spin
matrices).

The total Hamiltonian for FRW universe with a massless scalar
field, $\phi$, is found to be
\begin{equation}\label{ham}
   H = H_g + H_\phi ,
\end{equation}
where  $H_\phi = N\,p^2_\phi |p|^{-\frac{3}{2}}/2$, and where
$\phi$ and $p_\phi$ are canonical variables. Equation (\ref{ham})
satisfies the condition $ H\approx 0$ as it corresponds to the
{\it scalar constraint} of general relativity; the Gauss and
diffeomorphism constraints are equal zero (in the strong sense)
due to the choice of gauges.

Making use of (\ref{hol}) we calculate (\ref{hamL}) and get the
modified total Hamiltonian $H^{(\lambda)}_g$ corresponding to
(\ref{ham}) in the form \cite{Dzierzak:2009ip}
\begin{equation}\label{regH}
  H^{(\lambda)}/N = -\frac{3}{8\pi G \gamma^2}\;\frac{\sin^2(\lambda
\beta)}{\lambda^2}\;v + \frac{p_{\phi}^2}{2\, v} \approx 0,
\end{equation}
where
\begin{equation}\label{re1}
    \beta := \frac{c}{|p|^{1/2}},~~~v := |p|^{3/2}
\end{equation}
are the  canonical variables of the so called improved scheme, and
where $c$ and $p$ are canonical variables. The variable $\beta =
\gamma\dot{a}/a$ corresponds to the Hubble parameter $\dot{a}/a$,
and $v^{1/3} = a V_0^{1/3}$ is proportional to the scale factor
$a$. The relationship between the coordinate length $\mu$  (which
depends on $p$) and the physical length $\lambda$ (which is a
constant) reads:
    $\lambda = \mu \,|p|^{1/2} = \mu\, a \,V_0^{1/3}$.

At this stage, it should be emphasized that (\ref{regH}) presents
a purely classical Hamiltonian {\it modified} by the holonomy
(\ref{hol}), i.e. Eq. (\ref{regH}) includes no quantum physics.
Contrary, in the standard LQC (\ref{regH}) is called a
semi-classical or effective Hamiltonian and is interpreted to
include some `imprints' of quantization. In our nonstandard LQC
quantum physics enters the formalism only when quantizing the
algebra of observables (presented  in the subsequent section).

The complete Poisson bracket for the canonical variables
$(\beta,v,\phi,p_\phi)$ is defined to be
\begin{equation}\label{re2}
    \{\cdot,\cdot\}:= 4\pi G\gamma\;\bigg[ \frac{\partial \cdot}
    {\partial \beta} \frac{\partial \cdot}{\partial v} -
     \frac{\partial \cdot}{\partial v} \frac{\partial \cdot}{\partial \beta}\bigg] +
     \frac{\partial \cdot}{\partial \phi} \frac{\partial \cdot}{\partial p_\phi} -
     \frac{\partial \cdot}{\partial p_\phi} \frac{\partial \cdot}{\partial
     \phi}.
\end{equation}
The dynamics of a canonical variable $\xi$ is defined by
\begin{equation}\label{dyn}
    \dot{\xi} := \{\xi,H^{(\lambda)}\},~~~~~~\xi \in \{\beta,v,\phi,p_\phi\},
\end{equation}
where $\dot{\xi} := d\xi/d\tau$, and where $\tau$ is an evolution
parameter.

The dynamics in the  {\it physical} phase space,
$\mathcal{F}_{phys}^{(\lambda)}$, is defined by solutions to
(\ref{dyn}) satisfying the constraint $H^{(\lambda)}\approx 0$.
The solutions of (\ref{dyn}) ignoring the constraint  are in the
{\it kinematical} phase space.

\subsection{Elementary  observables}

A function, $\mathcal{O}$, defined on phase space   is a Dirac
{\it observable}  if
\begin{equation}\label{dirac}
\{\mathcal{O},H^{(\lambda)}\} \approx 0.
\end{equation}

Equation (\ref{dirac}) leads to \cite{Dzierzak:2009ip}
\begin{equation}\label{dir}
\frac{\sin(\lambda\beta)}{\lambda}\,\frac{\partial
\mathcal{O}}{\partial\beta} - v \cos(\lambda\beta)\,\frac{\partial
\mathcal{O}}{\partial v} - \frac{\kappa\,\textrm{sgn}(p_{\phi})}{4
\pi G}\,\frac{\partial \mathcal{O}}{\partial\phi} = 0,
\end{equation}
where $\kappa^2 := 4\pi G/3$.  Solution to (\ref{dir}) in
$\mathcal{F}_{phys}^{(\lambda)}$ is found to be
\cite{Dzierzak:2009ip}
\begin{equation}\label{obser1}
\mathcal{O}_1:= p_{\phi},~~~\mathcal{O}_2:= \phi -
\frac{\textrm{sgn}(p_\phi)}{3\kappa}\;\textrm{arth}\big(\cos(\lambda
\beta)\big).
\end{equation}
One may parameterize $\mathcal{F}_{phys}^{(\lambda)}$ by the {\it
elementary} observables $\mathcal{O}_1$ and $\mathcal{O}_2$, and
define the Poisson structure as follows \cite{Dzierzak:2009ip}
\begin{equation}\label{sym4}
\{\mathcal{O}_2,\mathcal{O}_1\}=
1,~~~~~~\{\cdot,\cdot\}:=\frac{\partial\cdot}{\partial
\mathcal{O}_2}\frac{\partial\cdot}{\partial \mathcal{O}_1} -
\frac{\partial\cdot}{\partial
\mathcal{O}_1}\frac{\partial\cdot}{\partial \mathcal{O}_2}.
\end{equation}
For simplicity we use the same notation for the Poisson bracket in
(\ref{sym4}) and  (\ref{re2}).

\subsection{Compound observables}

In what follows we consider functions on {\it physical} phase
space like the energy density and the volume, which describe
singularity aspects of our cosmological model. Considered
functions are expressed in terms of elementary observables and an
evolution parameter $\phi$. They become observables for each fixed
value of $\phi$, since in such case they are only functions of
observables \cite{Dzierzak:2009ip}.

An expression for the energy density $\rho$ of the scalar field
$\phi$ reads \cite{Dzierzak:2009ip}
\begin{equation}\label{rho2}
\rho(\phi,\lambda)=\frac{1}{2}\,\frac{1}{(\kappa\gamma\lambda)^2\,
    \cosh^2 3\kappa  (\phi- \mathcal{O}_2)}.
\end{equation}
The volume, a geometrical function, is found to be
\cite{Dzierzak:2009ip}
\begin{equation}\label{vol}
    v(\phi,\lambda) = \kappa\gamma\lambda\,
    |\mathcal{O}_1|\,\cosh3\kappa  (\phi-
    \mathcal{O}_2).
\end{equation}

\section{Quantization}

By quantization we mean: (i) finding a self-adjoint representation
of composite observables, and (ii) calculating spectra of
operators corresponding to the composite observables.

\subsection{Representation of elementary observables}

In what follows we use two representations  of the classical
algebra (\ref{sym4}). Namely,
\begin{equation}\label{quant1}
\mathcal{O}_1 \longrightarrow \widehat{\mathcal{O}}_1 f(x):=
-i\,\hbar\,\partial_x f(x),~~~~\mathcal{O}_2 \longrightarrow
\widehat{\mathcal{O}}_2 f(x):= \widehat{x} f(x) := x f(x),
\end{equation}
which leads to $[\widehat{\mathcal{O}}_1,\widehat{\mathcal{O}}_2]=
- i\,\hbar\,\mathbb I$, and
\begin{equation}\label{quant2}
\mathcal{O}_1 \longrightarrow \widehat{\mathcal{O}}_1 f(x):=
\widehat{x} f(x) := x f(x),~~~~\mathcal{O}_2 \longrightarrow
\widehat{\mathcal{O}}_2 f(x):= -i\,\hbar\,\partial_x f(x),
\end{equation}
that gives $[\widehat{\mathcal{O}}_1,\widehat{\mathcal{O}}_2]=
i\,\hbar\,\mathbb I$, where $x\in \mathbb{R}$. Due to the Stone$-$von
Neumann theorem all self-adjoint representations of the algebra
(\ref{sym4}) are unitarily equivalent to the representation
(\ref{quant1}) or (\ref{quant2}) defined on a suitable dense
subspace of $L^2(\mathbb R)$. In this sense our choice of representation
for (\ref{sym4}) is unique.

\subsection{Energy density}

The representation (\ref{quant2}) is  essentially self-adjoint on
on the dense subspace $D$ of the Hilbert space $L^2 [-r,r],\,$
where $r \in \mathbb R_+$, defined to be
\begin{equation}\label{den}
D:= \{f\in C^\infty [-r,r]\,|\, f^{(n)}(-r) = f^{(n)}(r),\, n \in
\{0\}\cup \mathbb N \},
\end{equation}
where $f^{(n)} := d^{\,n}f/dx^n$. The eigenvalue problem,
$\widehat{\mathcal{O}}_2 f_p = p\, f_p$, has the solution
\begin{equation}\label{sol1}
f_p (x) = (2r)^{-1/2} \exp(i x p /\hbar),~~~~p(k):= 2\pi\hbar
k/r,~~k \in \mathbb Z .
\end{equation}
The  spacing $ \square$ of neighboring eigenvalues
\begin{equation}\label{spa}
\square := p(k+1) - p(k) = 2\pi\hbar/r
\end{equation}
can be made as small as desired by making $r$ sufficiently large.
Thus, one may say that the  spectrum of $\widehat{\mathcal{O}}_2 $
is  continuous.

In the representation (\ref{quant2}) the energy density operator
reads
\begin{equation}\label{quant3}
 \widehat{\rho}:=\frac{1}{2}
    \frac{1}{(\kappa\gamma\lambda)^2\cosh^2 3\kappa(\phi+i\,\hbar\,\partial_x)}.
\end{equation}
Since $\widehat{\mathcal{O}}_2$ is {\it essentially} self-adjoint
on $\mathcal{F}_r := \{f_{p(k)}\}_{k \in \mathbb Z}$, we may apply the
spectral theorem to get
\begin{equation}\label{quant4}
\widehat{\rho}\,f_p = \rho (\phi,\lambda,p)\,f_p ,
\end{equation}
where
\begin{equation}\label{quant5}
\rho (\phi,\lambda,p) := \frac{1}{2} \frac{1}{(\kappa\gamma
\lambda)^2\cosh^2 3\kappa(\phi - p)},
\end{equation}
and where $\rho (\phi,\lambda,p)$ is the eigenvalue corresponding
to the eigenvector $f_p$.

Our results mean that classical (\ref{rho2}) and quantum
(\ref{quant5}) expressions for the energy density {\it coincide}.
It is clear that the maximum density, $\rho_{max}$, reads
\begin{equation}\label{maxio}
\rho_{max}(\lambda) = \frac{1}{2}\frac{1}{(\kappa\gamma
\lambda)^2}.
\end{equation}
Starting from the representation (\ref{quant1}), instead of
(\ref{quant2}), we would get the quantum model of the energy
density presented in \cite{Malkiewicz:2009zd}, which is equivalent
to the present one.

\subsection{Volume operator}

To define the quantum operator corresponding to $v$, we use
\begin{equation}\label{vol1}
v = |w|,~~~w :=
\kappa\gamma\lambda\;\mathcal{O}_1\;\cosh3\kappa(\phi-
\mathcal{O}_2).
\end{equation}
Thus, quantization of $v$ reduces to the  quantization problem of
$w$. Quantization of the latter may be done in a standard way as
follows
\begin{equation}\label{vol2}
\hat{w}\,f(x) :=
    \kappa\gamma\lambda\,\frac{1}{2}\,\bigg(
    \widehat{\mathcal{O}}_1\,\cosh3\kappa  (\phi-
    \widehat{\mathcal{O}}_2)+
     \cosh3\kappa  (\phi-
    \widehat{\mathcal{O}}_2)\;\widehat{\mathcal{O}}_1\bigg) f(x),
\end{equation}
where $f \in  L^2 (\mathbb R)$. For the elementary observables
$\mathcal{O}_1$ and $\mathcal{O}_2$ we use the  representation
(\ref{quant1}). An explicit form of the operator $\hat{w}$ is
\begin{equation}\label{repp1}
\hat{w}\,f(x)= i\,\frac{\kappa\gamma\lambda\hbar}{2}\bigg(
    2 \cosh3\kappa(\phi-x)\;\frac{d}{dx}
     -3\kappa\sinh3\kappa
    (\phi-x)\bigg)\,f(x).
\end{equation}
Considerations simplify if we take $f$ in the form
\begin{equation}\label{eq3}
f(x) :=  A\,e^{i h(x)}\, \cosh^{-1/2} 3 \kappa (\phi - x),
\end{equation}
where $A \in \mathbb R $, and where $h$ is a real-valued function.

\subsubsection{Eigenvalue problem}

Let us consider the eigenvalue problem for the operator $\hat{w}$
in the set of functions of the form (\ref{eq3}). We have
\begin{equation}\label{eq4}
\hat{w}\, f (x)= -\kappa\gamma\lambda\hbar
\cosh3\kappa(\phi-x)\frac{d h(x)}{dx}\, f(x) =: b\, f (x),
\end{equation}
where  $b \in \mathbb R $ is the eigenvalue of $\hat{w}$. One may verify
that a general form of $h$ satisfying (\ref{eq4}) is given by
\begin{equation}\label{eq55}
 h (x)=  \frac{2  b}{3\kappa^2
    \gamma\lambda\hbar}\arctan
    e^{3\kappa(\phi-x)}.
\end{equation}
Thus, a normalized $f_b $ satisfying (\ref{eq4}) reads
\begin{equation}\label{eq5}
f_b (x):= \frac{\sqrt{\frac{3\kappa}{\pi}}\exp\Big(i \frac{2
b}{3\kappa^2 \gamma\lambda\hbar}\arctan
    e^{3\kappa(\phi-x)}\Big)}{\cosh^{\frac{1}{2}}3\kappa(\phi-x)}.
\end{equation}

\subsubsection{Orthogonality}

Making use of (\ref{eq5}) gives
\begin{equation}
    \langle f_b|f_a\rangle= \frac{3\kappa}{\pi}\int_{-\infty}^{\infty}\frac{\exp\Big(i \frac{2
(a-b)}{3\kappa^2 \gamma\lambda\hbar}\arctan
    e^{3\kappa(\phi-x)}\Big)}{\cosh3\kappa(\phi-x)}~dx .
\end{equation}
We introduce a new variable $y=e^{3\kappa(\phi-x)}$ and have
\begin{equation}
   \langle f_b|f_a\rangle=\frac{2}{\pi}\int_{0}^{\infty}\frac{\exp\Big(i \frac{2
(a-b)}{3\kappa^2 \gamma\lambda\hbar}\arctan
    y\Big)}{1+y^2}~dy .
\end{equation}
Another substitution $\tan z =y$ leads to
\begin{equation}
   \langle f_b|f_a\rangle=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\exp\Big(i \frac{2
(a-b)}{3\kappa^2 \gamma\lambda\hbar}z\Big)~dz=-i\,
\frac{3\kappa^2\gamma\lambda\hbar}{\pi(a-b)}\exp\Big(i \,\frac{2
(a-b)}{3\kappa^2 \gamma\lambda\hbar}z\Big)\Big|_0^{\frac{\pi}{2}}
.
\end{equation}
It is clear that $\langle f_b|f_a\rangle = 0\;$ iff
\begin{equation}\label{ab}
    a-b = 6\kappa^2\gamma\lambda\hbar\,m = 8\pi
    G\gamma\lambda\hbar\,m,~~~~m\in \mathbb{Z} .
\end{equation}
Thus, the set $\mathcal{F}_b:=\{~f_a\;|\; a = b + 8\pi
G\gamma\lambda\hbar\,m;~m\in \mathbb Z;~b \in \mathbb R~\}$ is {\it
orthonormal}. Each subspace $\mathcal{F}_b \subset L^2(\mathbb R)$ spans
a pre-Hilbert space.  The completion of each $span
\;\mathcal{F}_b,~~\forall b \in \mathbb R$, in the norm of $L^2(\mathbb R)$
gives $L^2(\mathbb R)$.

\subsubsection{Self-adjointness}

Since $\langle f_b|\hat{w}f_a\rangle - \langle
\hat{w}f_b|f_a\rangle = (a-b)\langle f_b|f_a\rangle$, the operator
$\hat{w}$ is {\it symmetric} on $\mathcal{F}_b $ for any $b \in
\mathbb R$, because $\langle f_b|f_a\rangle=0$ for $a\neq b$ due to the
orthogonality of the set $\mathcal{F}_b$.

To examine the self-adjointness of the unbounded operator
$\hat{w}$, we first identify the deficiency subspaces,
$\mathcal{K_\pm}$,  of this operator \cite{RS,DS}
\begin{equation}\label{self1}
\mathcal{K_\pm}:= \{g_\pm \in D_b (\hat{w}^\ast)~|~\langle
g_\pm|(\hat{w} \pm i \mathbb I)f_a \rangle =0, \;\forall f_a \in D_b
(\hat{w})\},
\end{equation}
where $D_b (\hat{w}):=$ {\it span} $\mathcal{F}_b$, and where
$D_b(\hat{w}^{\ast}):=\{f\in L^2(\mathbb R) :~\exists !f^{\ast}~\langle
f^{\ast}|g\rangle=\langle f|\hat{w}g\rangle,~\forall g\in
D_b(\hat{w})\}$.

For each $f_a \in D_b(\hat{w})\subset L^2(\mathbb R)$ we have
\begin{equation}\label{self2}
0 = \langle g_\pm|(\hat{w} \pm i \mathbb I)f_a\rangle = (a \pm i)
\int_{-\infty}^\infty dx \;\overline{g_\pm (x)} f_a
(x)~~~~~\Longrightarrow~~~~~g_+ = 0 = g_- .
\end{equation}
Thus, the deficiency indices $n_\pm := dim [\mathcal{K}_\pm]$ of
$\hat{w}$ satisfy the relation: $n_+ = 0 = n_-$, which proves that
the operator $\hat{w}$ is {\it essentially} self-adjoint on
$D_b(\hat{w})$.

\subsubsection{Spectrum}

Due to the spectral theorem on self-adjoint operators
\cite{RS,DS}, we may carry out quantization of the volume as
follows
\begin{equation}\label{sp1}
 v = |w|~~~\longrightarrow~~~\hat{v} f_a :=  |a| f_a .
\end{equation}

A common feature of all $\mathcal{F}_b$ is the existence of the
minimum gap $\bigtriangleup  := 8\pi G\gamma\hbar\,\lambda\;$ in
the spectrum, which defines a  {\it quantum} of the volume.  Let
us discuss this issue in more detail. Let us denote the minimum
eigenvalue of $\hat{v}$ by $v_{min}$. One can verify that
$v_{min}=\textrm{min}\{b,\Delta-b\}$, where $b\in [0,\Delta[$. The
spectrum consists of the union of $\{v_{min}+n\Delta\}$ and
$\{-v_{min}+(n+1)\Delta\}$, where $n=0,1,\dots$ Only in two cases
these two subsets  are identical, namely when $\,v_{min}=0\,$ or
$\,v_{min}=\Delta/2\,$, for which the minimum gap $\Delta$ is a
{\it constant} gap between any two adjacent levels of the
spectrum. Otherwise, the gap  equals either $\,\Delta -
2v_{min}\,$ or $\,2v_{min}\,$, and the minimum gap is the smaller
one. One can verify that the case of any $b \in \mathbb R$ reduces to
the above case.

In the limit $\lambda \rightarrow 0$, corresponding to the
classical FRW model without the loop geometry modification, there
is no quantum of the volume.

 It results from (\ref{ab}) that for $b=0$ and $m=0$  the
minimum eigenvalue of $\hat{v}$ equals zero. It is a special case
that corresponds to the classical situation when $v=0$, which due
to (\ref{regH}) means that $p_\phi = 0$ so there is no classical
dynamics (for more details see \cite{Dzierzak:2009ip}). Thus, we
have a direct correspondence between classical and quantum levels
corresponding to this very special state. It is clear that all
other states describe  bouncing {\it dynamics}.

As the universe expands, a discrete spectrum of the volume
operator may favour a  {\it foamy} like structure which should
finally turn into a continuous spacetime. The quantum of a volume
may be used as a measure of a size, $\lambda_f$, of a spacetime
foam. One may speculate that $\lambda_f := \bigtriangleup^{1/3} =
\big(8\pi G\gamma\hbar\,\lambda\big)^{1/3}$.  Thus, an astro-cosmo
data that determine a size of  spacetime granularity may fix the
minimum length parameter $\lambda$ of LQC. That would enable
making an estimate of the critical matter density $\rho_{\max} =
1/2(\kappa \gamma \lambda)^2$ corresponding to the Big Bounce.

The granularity of volume should lead to the granularity of energy
of physical fields. We suggest, making use of the de Broglie
relation, that a specific  particle representing a quantum of
energy may have a momentum $p_i$ corresponding to its wavelength
$\lambda_i$ such that $p_i\,\lambda_i = \hbar$. The detection of
an ultrahigh energy particle with specific $p_i$ may be used to
determine $\lambda_i$, and consequently set the upper limit for
the fundamental length $\lambda_f$. The set of parameters
$\lambda_i$ (for a set of particles) may be treated further as
multiplicities of $\lambda_f$ in which case the greatest common
divisor of all $\lambda_i$ would set the lowest upper limit for
$\lambda_f$.

\subsubsection{Evolution}

The relation between eigenvectors corresponding to the same
eigenvalue for different values of the parameter $\phi$ reads:
\begin{equation}\label{tay1}
f_a^{\phi+\psi} = e^{\psi\partial_\phi}f_a^{\phi} =
e^{-i\frac{\psi}{\hbar} \widehat{\mathcal{O}}_1}f_a^{\phi} .
\end{equation}

One may verify that
\begin{equation}\label{da1}
    \hat{w}(\phi+\psi)=\cosh{(3\kappa\psi)}\hat{w}(\phi)
    +\frac{\sinh{(3\kappa\psi)}}
    {3\kappa}\partial_{\phi}\hat{w}(\phi),
\end{equation}
thus
\begin{eqnarray}\nonumber
    &&\langle f_b^{\phi}|\hat{w}(\phi+\psi)f_a^{\phi}\rangle =
    \langle f_b^{\phi}|\hat{w}(\phi)f_a^{\phi}\rangle\cosh{(3\kappa\psi)}+
    \frac{\sinh{(3\kappa\psi)}}{3\kappa}\langle f_b^{\phi}|\partial_{\phi}
    \hat{w}(\phi)f_a^{\phi}\rangle\\&&=a\cosh{(3\kappa\psi)}\,
    \delta_{ab}+(b-a)\frac{\sinh{(3\kappa\psi)}}{3\kappa}\langle
    f_b^{\phi}|\partial_xf_a^{\phi}\rangle .
\end{eqnarray}
Finally, an evolution of the  expectation value of the operator
$\hat{w}$ is found to be
\begin{equation}\label{da2}
    \langle
    f (\phi)|\hat{w}(\phi+\psi)f(\phi)\rangle = A \cosh3\kappa(\psi+B),
\end{equation}
where $f:=\sum\alpha_a f_a, \; f_a \in \mathcal{F}_b$.   One may
verify that
\begin{equation}
    A=\textrm{sgn}(X)\sqrt{X^2-Y^2},~~~~B=\frac{1}{6\kappa}\ln{\frac{X+Y}{X-Y}}~,
\end{equation}
where
\begin{equation}
    X:=\sum_{a}|\alpha_a|^2a,~~~~Y:=\sum_{a,~m}\frac{\bar{\alpha}_b\alpha_a-
    \bar{\alpha}_a\alpha_b}
   {i\pi}\frac{m(2a+6m\kappa^2\gamma\hbar\lambda)}{(2m-1)(2m+1)},
\end{equation}
and where $b = a + 6\kappa^2\gamma\lambda\hbar, \;\; b\in \mathbb R, \;
~m\in \mathbb{Z}$, and $|X|>|Y|$.

We can see that the evolution of the expectation value of the
operator $\hat{w}$ coincides with the classical expression
(\ref{vol1}).

\subsection{Energy density operator in the basis $\mathcal{F}_b$}

The operator $\hat{\rho}$ may be expressed in terms of the basis
vectors from $\mathcal{F}_b$. One may verify that
\begin{eqnarray}
\langle f_{a+m \bigtriangleup}|\hat{\rho} f_{a}\rangle= \left\{
\begin{array}{lll}
         \frac{-1}{8(\kappa\gamma\lambda)^2} & \mbox{if $\;\;m = 1$ or $-1$};\\
        \frac{1}{4(\kappa\gamma\lambda)^2} & \mbox{if $\;\;m = 0$};\\
        0 & \mbox{otherwise}\end{array} \right.
\end{eqnarray}
which leads to
\begin{equation}\label{nice}
\hat{\rho} f_{a}=
\frac{1}{4(\kappa\gamma\lambda)^2}f_{a}-\frac{1}{8(\kappa\gamma\lambda)^2}
f_{a-4 \bigtriangleup}-\frac{1}{8(\kappa\gamma\lambda)^2}f_{a+4
\bigtriangleup} ,
\end{equation}
where $\bigtriangleup  := 8\pi G\gamma\hbar\,\lambda\;$.
Therefore, $\hat{\rho}$ is bounded and does not commute with
$\hat{v}$. The latter is consistent with the Poisson bracket
relation for these observables, which reads
\begin{equation}\label{nonco}
\{w,\rho\}=  3 \kappa^2 \gamma \lambda \sinh 3\kappa  (\phi-
\mathcal{O}_2)\,\rho.
\end{equation}
Therefore, the operators $\hat{\rho}$ and $\hat{v}$ cannot have
common eigenfunctions so they provide two alternative ways of
describing the system, either in terms of the eigenfunctions of
$\hat{\rho}$ or $\hat{v}$.

\section{Summary and Conclusions}

Turning the big bang into the big bounce (BB) in the FRW universe
is due to the modification of the model at the {\it classical}
level by making use of the loop geometry. The modification is
parameterized by a {\it continuous} parameter  $\lambda $.

Each value of $\lambda$ specifies the critical energy density of
the scalar field corresponding to the BB. The spectrum of the
energy density operator is {\it bounded} and {\it continuous}.
Classical and quantum expressions for the minimum of the energy
density coincide. The spectrum of the volume operator,
parameterized by $\lambda$, is {\it bounded} from below and {\it
discrete}.  The expectation value of the volume operator coincide
with the classical expression. The results concerning the volume
operator may be extended to the area and the length operators
\cite{Dzierzak:2009ip}.

An {\it evolution} parameter, $\phi$, does not belong to the {\it
physical} phase space of our nonstandard LQC. Thus, it stays
classical during the quantization process
\cite{Malkiewicz:2009zd}. In the standard LQC, contrary to our
method, $\phi$ is a phase space variable so it must be quantized
\cite{Ashtekar:2006wn}. Being a quantum variable it may {\it
fluctuate} so its use as an evolution parameter at the quantum
level has poor interpretation.

The Hamiltonin {\it constraints} are treated differently in the
standard and nonstandard LQC at the classical level. In the former
case the Gauss and diffeomorphism constraints are turned into zero
by a suitable choice of gauges; the scalar constraint is solved
only at the quantum level.  In the latter case all the constraints
are classically solved leading to the {\it physical} phase space.
Thus, considered observables are  not kinematical, but physical.
This is why an evolution of geometrical functions like the volume
may be used for testing the singularity aspects of a given
cosmological model. The standard and our LQC methods give similar
results in the sense that they lead to  the Big Bounce transition.
However, our method is fully controlled analytically as it does
not require any numerical work.

We believe that our nonstandard LQC may be related with the
reduced phase space quantization of Loop Quantum Gravity (LQG)
(see \cite{Giesel:2007wn} and references therein). Finding the
correspondence may help  deriving LQC from LQG.

There exist results concerning the spectrum of the volume operator
obtained within LQG (see, e.g.
\cite{Ashtekar:1997fb,Meissner:2005mx}), but cannot be compared
easily with our results due to the lack of a direct correspondence
between LQG and LQC models.

As there is no specific choice of $\lambda$, the BB may occur at
any low and high densities \cite{Dzierzak:2008dy}. Finding
specific value of the parameter $\lambda$ (and energy scale
specific to the BB) is an open problem. It may happen, that the
value of the parameter $\lambda$ cannot be  determined, for some
reason, theoretically. Fortunately, there is a rapidly growing
number of data coming from observational cosmology that may be
useful in this context. In particular, the detection of the
primordial gravitational waves created at  the BB phase may bring
valuable information about this phase
\cite{Mielczarek:2008pf,Calcagni:2008ig,Grain:2009kw}. There
exists speculation that the foamy structure of spacetime may lead
to the dependence of the velocity of a photon on its energy. Such
dependance is weak, but may sum up to give a measurable effect in
the case of  photons travelling over cosmological distances across
the Universe \cite{GAC}. Presently, available data suggest that
such {\it dispersion} effects do not occur up to the energy scale
$5\times 10^{17}$ GeV \cite{Aharonian:2008kz}  so such effects may
be present, but at higher energies.

Various forms of  discreteness of spacetime underly many
approaches in fundamental physics. Just to name a few:
noncommutative geometry \cite{Heller:2006uq}, causal sets approach
\cite{Rideout:2008df}, gravitational Wilson loops
\cite{Hamber:2009uz}, Regge calculus \cite{Bahr:2009qc}, path
integral over geometries \cite{Ambjorn:2009ts}, spin foam model
\cite{Kaminski:2009fm}, and categories \cite{Baez:2009as}. The
discreteness may translate at the semi-classical level into a
foamy structure of space. Such expected property of spacetime
creates large activity in observational astrophysics and cosmology
(see, e.g. Lorentz and CPT violation \cite{Kostelecky:2008ts},
dispersion of cosmic photons \cite{AmelinoCamelia:2009pg},
electrons \cite{Galaverni:2008yj} and neutrinos
\cite{Ellis:2008fc}, birefringence effects \cite{Gleiser:2001rm}).
Our results concerning the physics of geometry at short distances
give some support to these approaches and expectations.

For more concluding remarks, connected with motivation for our
approach, we recommend reading the appendix section.

\begin{acknowledgments}
We are grateful to Piotr Dzier\.{z}ak for helpful discussions.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


The manuscript is prepared in a A4 paper, two coulumn, and single line spacing format.
A new paragraph should start 1 centimetre from the left margin, using 11-size, Garamond font type.
Acknowledgment, if any, should include the funding agency and the contract year.


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NZEDGE > News > Society > Politics and Economics > Jacinda Ardern On Universally Accessible Arts
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Jacinda Ardern On Universally Accessible Arts
New Zealand's Prime Minister and Minister for Arts, Culture and Heritage Jacinda Ardern says art should be for the many, not the few. The opinion piece features in Australia's ArtsHub, and was originally published at The Big Idea.
"Cultural engagement has multiple benefits. As Minister my priority is to make sure the three words in that title – Arts, Culture, Heritage become ingrained in people's brains and experiences and that their value to New Zealand society is far more widely recognised," Ardern says.
"The challenge, of course, is that the benefits of art, and of culture, are not always readily available to every New Zealander.
"They should be. Every one of us – be it an opera-loving elder in provincial Southland or a budding filmmaker in Mangere – deserves access to the diverse and varied talent that makes up our creative sector.
"I believe the arts and creativity are integral and inseparable parts of what it is to be human."
Original article by Jacinda Ardern, The Big Idea, ArtsHub, September 4, 2018.
Tags: Arts ArtsHub Big Idea (The) Jacinda Ardern
Brenda Wilt - 8:21 pm on September 11th, 2018
I totally agree with Jacinda Ardern - everyone should be exposed to the arts. Coming from a very lower/middle class family in Wellington, we were fortunate that due to my mothers love of music and the arts and her service to the arts community (the New Zealand Ballet Company, the New Zealand Opera Company, the Alex Lindsay String Orchestra) that we as children were exposed to music and art that our family could have never afforded to. The arts and culture should be available to everyone. I do have to lift up New Zealand for the amazing concerts in the park that gives all the opportunity to hear symphony, opera and other genres for free. We have nothing like that in Canada....
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